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THE LOGARITHMIC BRUNN-MINKOWSKI CONJECTURE
A complete and readable account
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HOW TO READ THIS.
This document proves one theorem: the logarithmic Brunn-Minkowski
inequality for symmetric convex bodies, in every dimension. It then
derives the two consequences the original paper advertises: the
symmetric L_p inequality, and the (B)-conjecture for even log-concave
measures.
Everything is stated and proved from the beginning. No step is
hidden. A few tools are imported from the research literature (the
real Monge-Ampere moment theorem of Berman-Berndtsson, a Hessian
estimate of Klartag, and standard elliptic regularity). Each imported
tool is stated precisely in the appendix, so you can see exactly what
is used and why.
The reader we have in mind can follow a proof line by line. If you
know what a convex set is, what volume is, and what a derivative is,
you can read this. Formulas use plain ASCII: R^n is n-dimensional
space, <x,u> is a dot product, |K| is the volume of K, and "subset"
is C. A superscript like h^t means a power.
A note on honesty. The deep analytic inputs (solving a real
Monge-Ampere equation, elliptic regularity) are genuinely deep. We
state them and use them; we do not pretend they are elementary. What
this document makes fully explicit is how they combine to give the
inequality. That combination is the content of the proof.
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PART 0. THE PROBLEM, IN PLAIN WORDS
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A convex body K in R^n is a compact convex set with nonempty
interior. If K = -K (the mirror image of every point is also in K)
we call it origin-symmetric. Volume is written |K|.
Classical Brunn-Minkowski. For convex bodies K, L and t in [0,1],
form the Minkowski average
(1-t)K + tL = { (1-t)x + t y : x in K, y in L }.
This is the set of all weighted averages of a point of K and a point
of L. Brunn-Minkowski says
|(1-t)K + tL| >= |K|^{1-t} |L|^t .
In one dimension a convex body is an interval, say [-a,a] and
[-b,b]. The average is the interval of half-width (1-t)a + t b, so
its length is 2((1-t)a+tb). The inequality is then exactly the
arithmetic-geometric mean inequality: (1-t)a + tb >= a^{1-t} b^t.
That is the whole content in one dimension. In higher dimensions it
is the statement that volume is a log-concave function of this
averaging. It is a cornerstone of convex geometry.
Why symmetric sets? The general (non-symmetric) log version is false;
the conjecture is made for origin-symmetric bodies. We prove it there,
in every dimension.
The refinement. For a unit vector u let
h_K(u) = max_{x in K} <x,u>
be the support function of K: the furthest K reaches in direction u.
A convex body is the intersection of the half-spaces
{ x : <x,u> <= h_K(u) } over all unit u, so h_K encodes K
completely.
Instead of averaging the sets linearly, average their support bounds
geometrically. For 0 <= t <= 1 the geometric mean of the bounds is
the function u |-> h_K(u)^{1-t} h_L(u)^t. This is usually not the
support function of any body. But we can still feed it into the
half-space construction. Define the Wulff body
W[f] = intersection over unit u of { x : <x,u> <= f(u) }.
For positive continuous f this is a genuine convex body (it contains
a small ball of radius min f and sits inside a ball of radius max f).
The logarithmic Brunn-Minkowski conjecture, proved here, is:
| W[ h_K^{1-t} h_L^t ] | >= |K|^{1-t} |L|^t . (BM)
Because the geometric mean is pointwise at most the arithmetic mean,
the Wulff body sits inside the Minkowski average:
W[ h_K^{1-t} h_L^t ] C (1-t)K + tL .
So (BM) asks for the same lower bound on volume but for a smaller
body. It is strictly stronger than classical Brunn-Minkowski. That is
why it was a conjecture, and why it is hard.
The difficulty in one sentence: the Wulff body is built from a
geometric average of support data, and volume does not behave simply
under geometric (as opposed to linear) averaging of that data. There
is no elementary inequality that computes the volume of the Wulff
body from |K| and |L|. The proof must go through analysis.
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PART 1. THE TOOLBOX
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Support function h_K and Wulff body W[f] as above. Three facts we use
constantly, read off from the half-space definition:
(monotone) f <= g implies W[f] C W[g];
(scaling) W[c f] = c W[f] for a constant c > 0;
(support) W[h_K] = K.
Probability measures. For a probability measure mu on R^n write
E_mu f = integral f dmu and Var_mu(f) = E_mu (f - E_mu f)^2.
If mu has a density we write dmu = rho(x) dx.
Log-concave measure. A nonnegative measure mu is log-concave if
mu((1-t)A + tB) >= mu(A)^{1-t} mu(B)^t
for compact A, B and t in (0,1). A density rho is log-concave if
rho = e^{-V} with V convex (V allowed to be +infinity, meaning
rho = 0 there). Even means mu(A) = mu(-A).
The objects we work with: a probability measure on the whole space
with a smooth, strictly positive, even density
dmu = Z^{-1} e^{-V(x)} dx, where V is even and strictly convex.
Moment coordinates. The analytic engine. Given such a mu, we look for
a change of variable x = grad phi(z), where phi is convex and
nu(z) = e^{-phi(z)} dz is a probability measure pushed forward to mu
by grad phi. The idea is optimal transport: grad phi is the transport
map, and it rewrites the general weight e^{-V} as a weight e^{-phi}
whose curvature tau = D^2 phi is a bounded, positive definite matrix
field. Everything downstream is done in the x variables with the
metric tau carried along. Part 5 constructs this change of variable.
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PART 2. THE MAIN THEOREM AND ITS CONSEQUENCES
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THEOREM (even log-Brunn-Minkowski). Let n >= 1. Let K, L be
origin-symmetric convex bodies in R^n. For every 0 <= t <= 1,
| W[ h_K^{1-t} h_L^t ] | >= |K|^{1-t} |L|^t . (BM)
COROLLARY (symmetric L_p Brunn-Minkowski). Let 0 < p < 1. With K, L
full-dimensional origin-symmetric, for 0 <= t <= 1,
| W[ ((1-t) h_K^p + t h_L^p)^{1/p} ] |^{p/n}
>= (1-t) |K|^{p/n} + t |L|^{p/n} .
COROLLARY (even log-concave measures; the (B)-conjecture). Let mu be
an even log-concave Radon measure on R^n. Then for origin-symmetric
convex bodies K, L and 0 <= t <= 1,
mu( W[ h_K^{1-t} h_L^t ] ) >= mu(K)^{1-t} mu(L)^t .
In particular the function t |-> mu(e^t K) is log-concave on R.
We prove the two corollaries now, assuming (BM). The theorem itself
occupies Parts 3-9.
PROOF of the L_p corollary from (BM).
At t = 0 or 1 the Wulff body is K or L and the formula is an
equality. Assume 0 < t < 1. Put
a = |K|^{1/n} > 0, b = |L|^{1/n} > 0,
K0 = K/a, L0 = L/b, so |K0| = |L0| = 1,
c = ((1-t) a^p + t b^p)^{1/p} > 0,
s = t b^p / c^p , so 0 < s < 1 and 1 - s = (1-t) a^p / c^p.
Since h_K = a h_{K0} and h_L = b h_{L0}, and the weighted power mean
of two positive numbers dominates their weighted geometric mean
(this is AM-GM applied to h_{K0}^p and h_{L0}^p):
((1-t) h_K^p + t h_L^p)^{1/p}
= c ((1-s) h_{K0}^p + s h_{L0}^p)^{1/p}
>= c h_{K0}^{1-s} h_{L0}^s .
Apply W, using scaling and monotonicity:
c W[ h_{K0}^{1-s} h_{L0}^s ] C W[ ((1-t) h_K^p + t h_L^p)^{1/p} ] .
By (BM) applied to K0, L0 (origin-symmetric, unit volume), the left
body has volume at least 1. Hence the right body has volume at least
c^n. Raise to the power p/n and use
c^p = (1-t)|K|^{p/n} + t|L|^{p/n}. This is the corollary. QED.
PROOF of the measure corollary from (BM).
Fix 0 < t < 1 and write D = W[h_K^{1-t} h_L^t]. If mu is the zero
measure the claim is immediate. Otherwise choose a closed ball C
centered at 0, large enough that K, L, D are all inside C and
0 < mu(C) < infinity (possible because mu is a nonzero Radon measure).
The probability measure
nu(A) = mu(A intersect C) / mu(C)
is even and log-concave: for compact A, B, convexity of C gives
(1-t)(A cap C) + t(B cap C) is inside ((1-t)A + tB) cap C, and the
defining inequality for mu applies. Since K, L, D are inside C, the
factor mu(C) cancels, so it is enough to prove the inequality for nu.
Let E be the affine hull of the support of nu. Evenness gives E = -E,
so E is a linear subspace. If dim E = 0 then nu = delta_0, all three
bodies contain 0, and we have equality. So let d = dim E >= 1 and
identify E with R^d.
Borell's theorem: an even log-concave probability on E whose support
spans E is an invertible affine image of a log-concave density on
R^d. Since the support spans, the affine map is invertible, so nu has
a log-concave density f on E. Evenness gives f(x) = f(-x) almost
everywhere; replacing f(x) by the geometric mean of f(x) and f(-x)
gives an even representative.
Set K_E = K intersect E and L_E = L intersect E. These are origin-
symmetric convex bodies in E, because K and L contain neighborhoods
of 0. Define their relative support functions h^E_{K_E}, h^E_{L_E}
using vectors v in E, and the relative Wulff body D_E in E from the
geometric mean of those relative supports. Let P_E be the orthogonal
projection onto E. For x in D_E and any unit u in R^n,
<x,u> = <x, P_E u> <= h^E_{K_E}(P_E u)^{1-t} h^E_{L_E}(P_E u)^t
<= h_K(u)^{1-t} h_L(u)^t ,
because the relative support is at most the full support. Hence
D_E C D intersect E . (rel)
Now apply (BM) in dimension d, then Saroglou's transfer in dimension
d, to the even log-concave density f. This gives
nu(D_E) >= nu(K_E)^{1-t} nu(L_E)^t .
Since nu is supported on E and D_E is inside D cap E,
nu(D) = nu(D cap E) >= nu(D_E) >= nu(K)^{1-t} nu(L)^t ,
which is the measure inequality. At t = 0, 1 the Wulff body is K or L.
This proves the first display.
For the (B) part: the support identity h_{e^a K} = e^a h_K gives
W[ h_{e^a K}^{1-t} h_{e^b K}^t ] = e^{(1-t)a + t b} K .
Apply the measure inequality to the two bodies e^a K and e^b K:
mu( e^{(1-t)a+t b} K ) >= mu(e^a K)^{1-t} mu(e^b K)^t .
This is exactly log-concavity of t |-> mu(e^t K). QED.
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PART 3. THE PROOF STRATEGY, IN THREE STEPS
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Here is the whole plan before the details, so you always know where
you are.
Step A (reduction, Part 4). Approximate K and L by a symmetric
polytope whose sides are slabs |<p_i,x>| <= h_i. Replace the sharp
cutoff "inside the slab" by a smooth penalty ((<p_i,x>/h_i)^q) and
add a tiny Gaussian. For this smooth object the volume is an integral
Z(t) depending on the slab widths h_i(t). The second derivative of
log Z(t) is a variance minus an expected second derivative. So the
volume inequality would follow from ONE variance inequality for a
single smooth potential. That inequality is Proposition (VAR) below.
Step B (the variance inequality, Parts 5-9). Prove (VAR). This is the
analytic heart. In moment coordinates the potential becomes a convex
phi and we get a bounded "curvature" tau = D^2 phi. A certain
differential operator A is built from tau. A Hessian identity (Part 6)
identifies a natural quadratic form. A density lemma (Part 7) says the
test functions used by A are dense among even centered L^2 functions.
A tensor/Bochner estimate (Part 8) gives a sum-of-squares bound. Two
Cauchy-Schwarz steps (Part 9) turn this into (VAR).
Step C (limits, end of Part 4). Let q -> infinity (sharp slabs),
eps -> 0 (remove the Gaussian), and the slab directions become dense
(recover the Wulff body). Continuity of volume handles the limits.
The rest of this document is exactly these three steps, spelled out.
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PART 4. STEP A: FROM VOLUME TO A VARIANCE BOUND
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The variance proposition.
PROPOSITION (VAR). Let n >= 2, N >= 1, q >= 2 an even integer, eps > 0.
Let p_1,...,p_N be nonzero vectors in R^n and h_1,...,h_N > 0. Put
psi_i(x) = (<p_i,x>/h_i)^q ,
V(x) = (eps/2)|x|^2 + sum_i psi_i(x) ,
dmu = Z^{-1} e^{-V(x)} dx , Z = integral e^{-V} .
Then for all real b_1,...,b_N,
Var_mu( sum_i b_i psi_i ) <= sum_i b_i^2 E_mu psi_i . (VAR)
The Gaussian term makes Z finite even when the p_i do not span R^n.
All polynomial moments are finite. We prove (VAR) in Parts 5-9; its
dual form is
sum_i (E_mu u psi_i)^2 / (E_mu psi_i) <= E_mu u^2
for every centered even u in L^2(mu). (VAR')
The denominators are positive because p_i != 0 and mu has a positive
density. (VAR') implies (VAR): apply it to
u = sum_i b_i psi_i, subtract the mean (this does not change
variances because of the centering step in the Cauchy-Schwarz below).
Why (VAR) gives volume. Suppose the p_i span R^n and the widths are
positive. For t in [0,1] set
h_i(t) = (h_i^0)^{1-t} (h_i^1)^t ,
P(t) = { x : |<p_i,x>| <= h_i(t) for all i } .
The P(t) are bounded polytopes with nonempty interior. Use the widths
h_i(t) in the potential of (VAR) and call the resulting integral
Z_{q,eps}(t). Put b_i = -q log(h_i^1 / h_i^0). Differentiating
psi_i in t:
d/dt psi_i = b_i psi_i , d^2/dt^2 psi_i = b_i^2 psi_i .
So V_t has derivative Vdot = sum_i b_i psi_i and second derivative
Vddot = sum_i b_i^2 psi_i. Twice differentiating log Z (derivatives
pass under the integral because a Gaussian times a polynomial bounds
the differentiated integrands), and using the standard identity
d^2/dt^2 log Z = Var_mu(Vdot) - E_mu(Vddot), we get
d^2/dt^2 log Z_{q,eps}(t)
= Var_{mu_t}( sum_i b_i psi_i ) - E_{mu_t}( sum_i b_i^2 psi_i )
<= 0 by (VAR).
Hence t |-> log Z_{q,eps}(t) is concave, i.e.
Z_{q,eps}(lambda) >= Z_{q,eps}(0)^{1-lambda} Z_{q,eps}(1)^lambda .
Now take the limits.
q -> infinity (even integers). For fixed t the integrands converge
almost everywhere to e^{-eps|x|^2/2} 1_{P(t)}. The exceptional set is
a finite union of hyperplanes <p_i,x> = +-h_i(t), which has measure
zero. The Gaussian e^{-eps|x|^2/2} dominates everything, so dominated
convergence gives Z_{q,eps}(t) -> integral_{P(t)} e^{-eps|x|^2/2} dx.
Applying this at t = 0, lambda, 1 preserves the concavity.
eps -> 0. Monotone convergence removes the Gaussian and gives
|P(lambda)| >= |P(0)|^{1-lambda} |P(1)|^lambda . (slabs)
Dense directions. Let K, L be the given symmetric bodies. Choose
nested finite direction sets D_m on the unit sphere, each containing
the coordinate vectors, with union dense in the sphere. Set
P_m(t) = { x : |<p,x>| <= h_K(p)^{1-t} h_L(p)^t for all p in D_m } .
The P_m(t) decrease in m and all lie inside the bounded body P_1(t).
By continuity of the support functions,
intersection_m P_m(t) = W[ h_K^{1-t} h_L^t ] .
(One direction: a point satisfying the dense inequalities satisfies
all of them by continuity. The other direction: the Wulff body is
defined by all inequalities, and by evenness the absolute-value form
matches.) At t = 0, 1 these intersections are K and L by the
half-space description of a convex body. Continuity of Lebesgue
measure from above, applied at t = 0, lambda, 1 in (slabs), gives
(BM). This finishes Step A and Step C.
The one-dimensional case n = 1 is separate and trivial. Write
K = [-a,a], L = [-b,b] with a, b > 0. The Wulff interpolation is the
interval with radius a^{1-lambda} b^lambda, whose length is
(2a)^{1-lambda}(2b)^lambda, which is equality in (BM).
So everything now rests on (VAR). Parts 5-9 prove it.
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PART 5. STEP B1: MOMENT COORDINATES
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We want to replace the awkward weight e^{-V} by a weight e^{-phi}
whose curvature D^2 phi is a nice bounded matrix field, moving a
general even strictly convex V into a single canonical shape.
LEMMA (moment coordinates). Let n >= 2, eps > 0, and let
V in C^infinity(R^n) be even with D^2 V >= eps I (as quadratic
forms). Set Z = integral e^{-V} dx and dmu = Z^{-1} e^{-V} dx. There
is an even phi in C^infinity(R^n) such that
nu = e^{-phi} dz is a probability measure,
grad phi is a diffeomorphism of R^n onto R^n,
(grad phi)_# nu = mu (grad phi pushes nu to mu),
0 < D^2 phi <= (4/eps) I (positive definite, bounded above),
log det D^2 phi = V(grad phi) + log Z - phi ,
phi(z) >= c|z| - b for some c > 0, b < infinity.
(The lower bound D^2 phi > 0 is pointwise positivity, not a uniform
lower bound.) Write x = grad phi(z). The last-but-one line is the
pushforward equation: the change of variables x = grad phi turns
e^{-phi(z)} dz into e^{-V(x)} dx / Z.
Idea of the proof (the full argument, with the imported tools named).
1. Compact approximation. For R >= 2 let B_R be the ball of radius R.
Apply the Berman-Berndtsson real Monge-Ampere moment theorem to
the ball B_R and the density e^{-V}/Z_R on it. It gives a smooth
convex potential phi_R with grad phi_R : R^n -> B_R a
diffeomorphism and det D^2 phi_R = Z_R exp(V(grad phi_R) - phi_R).
Integrating through the map shows e^{-phi_R} is a probability
pushing to the normalized restriction of mu to B_R. This is the
whole reason for the compact approximation: the moment theorem is
cleanest on a bounded target.
2. Uniform curvature bound. Klartag's Hessian estimate, applied on
the smooth convex ball, gives 0 < D^2 phi_R <= (4/eps) I with a
constant independent of R (here we use D^2 V >= eps I).
3. Symmetry and coercivity. phi_R, normalized so its minimum sits at
0, is even (reflection gives a second solution with the same
minimum, and the solution is unique up to translation). Supporting
planes at the preimages of +-(R/2)e_i give phi_R(z) >= c|z| - b,
so e^{-phi_R} has an exponential tail and integration by parts is
legal. In particular E_{nu_R} z.grad phi_R = n (integrate the
divergence of z e^{-phi_R}).
4. Bounds independent of R. With g_R = phi_R - phi_R(0) >= 0 one gets
a two-sided bound c|y| - n <= g_R(y) <= (C/2)|y|^2, which controls
the normalization constants and gives a single integrable
exponential majorant for all the e^{-phi_R}.
5. Compactness and limit. On each fixed ball the estimates bound
phi_R and grad phi_R uniformly, and det D^2 phi_R >= const > 0
bounds D^2 phi_R below. So D^2 phi_R is uniformly elliptic there.
The spectral-function trick (log det is concave and uniformly
elliptic after smoothing the logarithm off a compact spectral
interval) plus Evans-Krylov regularity and then Schauder estimates
give uniform C^{k,alpha} bounds on compact sets. A diagonal
subsequence converges locally smoothly to an even phi with the same
properties and the moment equation. The common exponential
majorant passes the pushforward identity to the limit: for bounded
continuous h,
integral h(grad phi) e^{-phi}
= lim_R integral h(grad phi_R) e^{-phi_R}
= lim_R integral h dmu_R
= integral h dmu .
The Hessian D^2 phi is positive definite, so grad phi is injective
(the fundamental theorem of calculus along segments gives strict
monotonicity). Its image is dense (a missed ball would carry zero
pushforward mass, contradicting positive density of mu). Density
plus a supporting-plane argument shows grad phi is onto. This
completes the lemma. QED (with Steps 1-5 as described).
The point of all this: after the change of variable, the curvature
tau = D^2 phi is a smooth, even, positive definite, upper-bounded
(by (4/eps) I) matrix field, and its determinant is tied to V by the
moment equation. All of Parts 6-9 are computations in these
coordinates.
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PART 6. STEP B2: THE OPERATOR A AND THE HESSIAN IDENTITY
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Fix the objects of Lemma (moment coordinates):
x = grad phi(z), tau = D_z^2 phi(z), M = tau^{-1},
and interpret any matrix field in either the z or the x variable
through the map, denoting expectation under either measure by E.
The weighted divergence. For a vector field W in x-coordinates define
delta_mu W = sum_j ( V_{x_j} W_j - dW_j/dx_j ) .
On a matrix, delta_mu acts row by row and gives a vector. Weighted
integration by parts (the adjoint of the gradient in L^2(mu)) gives,
for a smooth row field P_{ak} and compact test g,
E g (delta_mu(P tau))_a = E (d_k g) P_{ak} = E (d_i g) tau_{ik} P_{ak} .
delta_mu(tau grad f) = x . grad_x f - tau : D_x^2 f .
Here tau : Q = trace(tau^T Q) is the matrix contraction. Define
A f = delta_mu(tau grad_x f) = x . grad_x f - tau : D_x^2 f .
(The chain rule and the moment equation give the middle identity; this
is Fathi's moment-map construction of a Stein kernel: tau expresses
integration by parts against x through E x_j g = E tau_{jk} dg/dx_k.)
Facts about A, verified directly from the formula:
A 1 = 0 ,
A x_j = x_j (so A fixes the coordinate functions) ,
A is formally symmetric:
E g A f = E grad_x g . tau grad_x f (form)
for compact smooth f, g. In particular E A k = 0 for compact k,
and A(A-1) kills every affine function.
The compact test fields. For f in C_c^infinity(R_x^n) put
h = D_x^2 f , P = tau h , B = tau h tau ,
W = delta_mu B (a vector, row-wise divergence) ,
u = delta_mu W .
All these fields have compact support in both variables (the compact
support is preserved by the inverse diffeomorphism grad phi). Two
identities tie them to A and to the gradient:
W = tau grad_x( (A-1) f ) , u = A(A-1) f . (fields)
(The second follows from the first by applying delta_mu; the first is
a direct computation using full symmetry of D_z^3 phi, in which the
first-order terms from x.grad f and -f cancel.)
The Hessian identity. This is the engine. Two compact integrations by
parts using (form) and (fields) give
E (Af)(A-1)f
= E grad_x f . tau grad_x (A-1)f
= E grad_x f . W
= E h : B
= E || tau^{1/2} D_x^2 f tau^{1/2} ||_HS^2 . (hess)
Here ||Q||_HS^2 = trace(Q^T Q). The last step uses B = tau h tau and
symmetry of h. Equation (hess) is a transport-Hessian Bochner formula;
it says the L^2 pairing of the tests A(A-1)f equals a manifestly
nonnegative Hessian energy. Part 7 uses (hess) to show these tests
are dense; Part 8 gets the other half of the story from a Bochner
identity for vector fields.
That is as far as we go this part. Next: the density lemma.
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PART 7. STEP B3: THE TESTS ARE DENSE
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We must know that the functions u = A(A-1)f, for even compact smooth
f, are plentiful: dense among centered even L^2(mu) functions. That is
what lets us move from "the variance inequality holds for these nice
tests" to "it holds for everything", which is exactly the step used at
the end of Part 9.
Throughout, Gamma(f,h) = grad f . tau grad h, Gamma(f) = Gamma(f,f),
and we use the integration formula and the Hessian identity (hess)
from Part 6. The integration formula extends to the case where one
factor is smooth and the other compactly supported (insert a cutoff
around the support). In particular E A k = 0 for every compact smooth
k. Since A preserves evenness, the range of A(A-1) on even f consists
of centered (mean-zero) even functions.
LEMMA (density). Assume A has the integration formula and satisfies
(hess). Then
closure of { A(A-1) f : f even, smooth, compactly supported }
in L^2(mu) = { g in L^2(mu) : g even, E g = 0 } .
Proof. Let g be a centered even L^2(mu) function orthogonal to the
range. It is also orthogonal to A(A-1)f for odd f, because that image
is odd and g is even. Formal symmetry of P = A(A-1) then gives
P g = 0 in the sense of distributions.
(Concretely: if P^t is the transpose in the flat Lebesgue measure and
dmu = rho dx with rho smooth positive, then weighted formal symmetry
gives P^t(rho g) = rho P g in distributions; orthogonality makes the
left side vanish. The principal symbol of P is (xi . tau(x) xi)^2,
positive for every nonzero xi because tau is positive definite, so P
is elliptic.)
Interior elliptic regularity then makes g smooth. Here it is enough
that the density is smooth and positive on compact sets; tau need only
be positive there.
Now split P g = 0 into a gainful form using cutoffs. Choose eta_r
smooth with 0 <= eta_r <= 1, equal to 1 on the ball B_r and 0 outside
B_{2r}, with |grad eta_r| <= C/r and |D^2 eta_r| <= C/r^2 (r >= 1).
The bound tau <= (4/eps)I gives constants B, K independent of r with
Gamma(eta_r)^{1/2} <= B/r , |A eta_r| <= K . (cutoff)
Put v = Ag (smooth), G = ||g||_2 (finite by hypothesis),
X_r = || eta_r^2 v ||_2, and Y_r^2 = E[ eta_r^2 Gamma(g) ] (finite,
compact support). Testing P g = 0 against eta_r^4 g gives
E[ v (A-1)(eta_r^4 g) ] = 0. The product rule A(ab) = a Ab + b Aa
- 2 Gamma(a,b) expands (A-1)(eta_r^4 g) exactly, and Cauchy-Schwarz
plus (cutoff) gives
X_r <= a G + (8B/r) Y_r ,
Y_r^2 <= G X_r + (2B/r) G Y_r , a = 1 + 4K + 12B^2 . (estimates)
Combine them: Y_r^2 <= a G^2 + 10 B G Y_r, so both Y_r and X_r are
bounded uniformly in r. Letting r exhaust the space proves v = Ag is
in L^2(mu). Now set
w = g - v in L^2(mu), A w = 0 , A v = v .
The harmonic part is constant. Test A w = 0 against eta_r^2 w:
( E[ eta_r^2 Gamma(w) ] )^{1/2} <= (2B/r) ||w||_2 .
On a fixed ball let r -> infinity: Gamma(w) = 0 there. Strict
positivity of tau makes w constant.
The eigenvalue-one part is affine. The same test of A v = v gives,
with Z_r^2 = E[eta_r^2 Gamma(v)],
Z_r^2 <= ||v||_2^2 + (2B/r)||v||_2 Z_r, hence E Gamma(v) < inf.
For f_r = eta_r v (now in C_c^infinity), the product rule shows
f_r -> v and A f_r -> v in L^2(mu). (approximation)
Apply (hess) to f_r:
E || tau^{1/2} D^2 f_r tau^{1/2} ||_HS^2
= ||A f_r||_2^2 - E[ f_r A f_r ] -> 0 .
On any fixed ball f_r = v for all large r, so D^2 v = 0 there:
v is affine. Apply A v = v to an affine function c + L.x; since
A1 = 0 and A x_j = x_j, this forces c = 0, so v = L.x is linear. But
g is even and w is constant, so v = g - w is even; an even affine
function is constant. Hence v = 0 and g = w is constant, and E g = 0
gives g = 0.
So the only centered even L^2 function orthogonal to the range is 0,
which proves density. QED.
============================================================
PART 8. STEP B4: THE TENSOR (BOCHNER) ESTIMATE
============================================================
We have a nonnegative Hessian energy in (hess). We also need control
of the gradient energy of the test fields. The next lemma bounds both,
in moment coordinates, by E u^2. It is the only place the moment
equation is differentiated to second order; homogeneity of the psi_i
enters only later (Part 9).
LEMMA (tensor estimate). Let dmu = Z^{-1} e^{-V} dx and
dnu = e^{-phi} dz be smooth probability measures on R^n. Suppose
D_z^2 phi is positive definite and x = grad phi(z) is a smooth
diffeomorphism pushing nu to mu. For compact smooth f in the x
variable put tau = D_z^2 phi, M = tau^{-1}, H = D_x^2 V,
B = tau (D_x^2 f) tau, W = delta_mu B, u = delta_mu W (all pulled to
either variable by the map). Then
E tr( (D_x W)^2 ) >= E tr( M B H B ) , (tensor1)
E u^2 >= E tr( M B H B ) + E W^T H W . (tensor2)
Here (D_x W)_{ij} = dW_i/dx_j and the square in (tensor1) is a
matrix square.
Proof. First, the Bochner identity for vector fields. For any such W
with u = delta_mu W, the product rule and adjunction give
E u^2 = E W_i d_i u
= E W^T H W + E (dW_j/dx_i)(dW_i/dx_j)
= E W^T H W + E tr( (D_x W)^2 ) . (bochner)
(Computation: d_i u = H_{ij} W_j + (V_{x_j} - d_j) d_i W_j. Pairing
with W_i, the first summand gives E W^T H W. The second, by adjunction
((V_{x_j} - d_j) is adjoint to d_j), gives E (d_j W_i)(d_i W_j) =
E tr((D_x W)^2).) So (tensor2) will follow from (bochner) and
(tensor1). We prove (tensor1).
Set h = D_x^2 f, P = tau h (from Part 6), and let (C_i)_{kl} = C_{ikl}
where C = D_z^3 phi. Recall the moment equation
V(x(z)) = phi(z) + log det tau(z) - log Z .
One derivative gives tau_{ik} V_{x_k} = x_i + c_i with c_i =
tr(M C_i). Differentiating again and recognizing the divergence of
M C_i (using dM = -M (d tau) M and full symmetry of D_z^4 phi) gives
(tau H tau - tau)_{ij}
= (d_d - x_d)( M_{dk} C_{ijk} ) - tr( M C_i M C_j ) . (twice)
For the other expansion, the commutator [d_l, d_k^*] = tau_{kl}
(of the x-gradient and its adjoint) gives
D_z W = B + R , R_{al} = d_k^* d_l P_{ak} .
Both D_z W and B are symmetric, so R is symmetric. Since D_x W = (B+R)M
and M B M = h, expansion and one integration by parts give
E tr( (D_x W)^2 )
= E tr(M B M B) + 2 E (d_k h_{al})(d_l P_{ak})
+ E tr(M R M R) . (trace-exp)
The last term is a Hilbert-Schmidt norm, tr(MRMR) =
|| M^{1/2} R M^{1/2} ||_HS^2 >= 0. On the right of (tensor1) the
coefficient of tau H tau in the contraction is Q = M B M B M = h tau h.
Substituting (twice) and integrating its divergence term by parts:
E tr(M B H B)
= E tr(M B M B)
- E (d_d Q_{ij}) M_{dk} C_{ijk}
- E Q_{ij} tr(M C_i M C_j) . (H-exp)
The common first term cancels against (trace-exp). What is left is a
comparison of the last two contracted terms with the cross term in
(trace-exp). We do the comparison in coordinates making tau = I at the
point under consideration; this is legitimate because every contracted
scalar below is invariant under a constant change of variable
z -> T z, x -> T^{-T} x (each z-index transforms covariantly, the
measure is unchanged), and the integrations by parts are already done
(no moving frame is differentiated).
In these normalized coordinates (tau = I at the point), put
S_{ijk} = f_{x_i x_j x_k} (totally symmetric) ,
J_{ijk} = h_{ia} C_{ajk} (symmetric in the last two indices),
and J^{(12)}_{ijk} = J_{jik}. Writing the Euclidean tensor norm and
inner product in this frame, the three remaining contractions are
2 (d_k h_{al})(d_l P_{ak}) = 2||S||^2 + 2<S,J> , (n1)
(d_d Q_{ij}) C_{ijd} = 2<S,J> + <J, J^{(12)}> , (n2)
Q_{ij} tr(C_i C_j) = ||J||^2 . (n3)
(For (n1): d_k h_{al} = S_{alk} and d_l P_{ak} = C_{ail} h_{ik} +
S_{akl}. For (n2): the two copies of <S,J> come from differentiating
the two outer factors h in Q = h tau h, and the middle-factor
derivative gives sum_d tr(h C_d h C_d) = <J, J^{(12)}>. For (n3):
expand Q = h^2 and use full symmetry of C.)
So, apart from the nonnegative R term, the integrand
(trace-exp) minus (H-exp) is
2||S||^2 + 4<S,J> + ||J||^2 + <J, J^{(12)}> .
Now symmetrize J: (Sym J)_{ijk} = (J_{ijk} + J_{jik} + J_{kij})/3.
Its norm satisfies ||Sym J||^2 = (||J||^2 + 2<J,J^{(12)}>)/3 and
<S, Sym J> = <S,J>. Therefore the displayed integrand equals
2 || S + Sym J ||^2 + (1/6) || J - J^{(12)} ||^2 >= 0 . (squares)
This proves (tensor1) pointwise (the step proving an inequality between
the integrands of (trace-exp) and (H-exp) uses exactly (squares));
combining with (bochner) proves (tensor2). QED.
One sentence on why this works: the Hessian energy of the test and the
contraction E tr(M B H B) differ by a perfect divergence minus terms
that reassemble into a sum of squares, so the energy dominates the
contraction. That domination is what Part 9 spends.
============================================================
PART 9. STEP B5: THE VARIANCE BOUND
============================================================
Now assemble. All expectations are under the probability mu of
Proposition (VAR). Write
H_i = D_x^2 psi_i , a_i = E psi_i > 0 .
Each H_i is positive semidefinite (a Hessian of a convex power), and
a_i > 0 since p_i != 0 and mu has a positive density. Homogeneity of
psi_i = (<p_i,x>/h_i)^q and Euler's identity give
x . grad psi_i = q psi_i , H_i x = (q-1) grad psi_i . (homog)
The adjoint identity also gives m_i := E tr(tau H_i) = q a_i. (Justify
it on the noncompact functions by applying the integration formula
E x_j k = E tau_{jk} d_k k to k = eta_R d_j psi_i and letting R go to
infinity; the boundary term is O(R^{-1}) E[ |grad psi_i| 1_{|x|>R} ],
which vanishes because tau is bounded and polynomial moments are
finite.)
Take an even f in C_c^infinity and set
B = tau (D_x^2 f) tau , W = delta_mu B , u = delta_mu W
= A(A-1) f , M = tau^{-1} .
Apply the test pairing (Part 6) with F = psi_i:
d_i := E u psi_i = E W . grad psi_i = E tr(B H_i) . (di)
Two Cauchy-Schwarz estimates bound these same numbers d_i.
First: Hilbert-Schmidt Cauchy-Schwarz (including integration) applied
to the matrix fields H_i^{1/2} tau^{1/2} and H_i^{1/2} B tau^{-1/2},
whose pairing is tr(H_i B):
d_i^2 <= m_i E tr( M B H_i B ) . (CS1)
This needs no positivity of B.
Second: from (homog)-(di), E x^T H_i x = (q-1) m_i and
E x^T H_i W = (q-1) d_i. Cauchy-Schwarz for the positive semidefinite
form (X,Y) |-> E X^T H_i Y therefore yields
E W^T H_i W >= (q-1) d_i^2 / m_i . (CS2)
Now apply the tensor lemma (Part 8) to H = D^2 V = eps I + sum_i H_i:
E u^2 >= E tr(M B H B) + E W^T H W
>= sum_i ( E tr(M B H_i B) + E W^T H_i W ) (drop eps I)
>= sum_i ( d_i^2/m_i + (q-1) d_i^2/m_i ) (CS1, CS2)
= sum_i q d_i^2 / m_i = sum_i d_i^2 / a_i . (dual)
The dropped terms are nonnegative because M is positive definite and B
is symmetric, and we used m_i = q a_i. This is (VAR') for the dense
class of double divergences u = A(A-1)f.
Finally extend to all centered even u. By the density lemma (Part 7)
these u are dense in the centered even subspace of L^2(mu). Each
psi_i is in L^2(mu), so the linear functionals u |-> E u psi_i are
continuous; inequality (dual) survives the limit. So (VAR') holds for
every centered even u in L^2(mu).
Take F = sum_i b_i psi_i and u = F - E F (centered and even). Then
||u||_2^2 = sum_i b_i d_i (with d_i = E u psi_i), and
||u||_2^2
<= ( sum_i b_i^2 a_i )^{1/2} ( sum_i d_i^2 / a_i )^{1/2}
<= ( sum_i b_i^2 a_i )^{1/2} ||u||_2 .
If ||u||_2 = 0 there is nothing to prove; otherwise divide and square.
This is exactly (VAR). Proposition (VAR) is proved, and with it, by
Part 4, the even log-Brunn-Minkowski inequality (BM).
============================================================
PART 10. PUTTING IT TOGETHER, AND WHAT WAS USED
============================================================
The chain is now complete:
- (BM) follows from (VAR) via the smooth slab reduction and three
limits (q -> infinity, eps -> 0, dense directions). Part 4.
- (VAR) reduces to (VAR') and to the operator A attached to moment
coordinates. Part 5 builds those coordinates and the curvature tau.
- The Hessian identity (hess) turns the test pairing into a Hessian
energy. Part 6.
- The density lemma says the tests A(A-1)f, f even compact smooth,
are dense in centered even L^2(mu). Part 7.
- The tensor lemma dominates the Hessian energy and the gradient
energy by E u^2, via a Bochner identity plus a sum-of-squares
tensor computation. Part 8.
- Two Cauchy-Schwarz estimates and the density lemma give (VAR'). Part 9.
The whole proof therefore comes down to one mechanism: in moment
coordinates, the natural differential operator A satisfies a Hessian
identity, and the difference between its energy and the moments of the
potential is a sum of squares. That is the non-obvious fact, and it is
where the real content of the theorem lives. The imported regularity
tools (Monge-Ampere existence, elliptic regularity) are exactly what is
needed to make tau a legitimate object; they are listed in the
appendix.
Why it is true, in one paragraph. Brunn-Minkowski usually follows from
a Brascamp-Lieb-type variance bound for log-concave measures. The
logarithmic version replaces the linear Brunn-Minkowski sum by the
geometric-mean Wulff body, which is not additive. The trick is to
choose coordinates adapted to the specific measure (moment coordinates)
so that the "curvature" of the weight is a bounded positive matrix tau.
In those coordinates the relevant variance is controlled by a Bochner
identity plus a tensor inequality that is literally a squared norm.
That is the sum of squares. The density lemma removes the boundary
obstructions (only affine functions survive), and evenness plus
centering kill those. Put differently: the geometry of the Wulff body
is analyzed through the analysis of a single diffusion, and the
diffusion's generator has positive Hessian energy by a Bochner formula.
============================================================
APPENDIX. THE IMPORTED TOOLS, STATED PRECISELY
============================================================
These are the results taken from the literature. Each is stated in the
form used. All are standard inside convex geometry and PDE; none is
hidden.
A1. Real Monge-Ampere moment theorem (Berman-Berndtsson). Let B be a
smooth bounded open convex set and let rho be a positive smooth
density on B with a normalization. There exist a convex potential
phi and a constant such that grad phi maps R^n diffeomorphically onto
B, and with x = grad phi(z),
det D^2 phi(z) = (a partition factor) exp( V(x) - phi(z) ) .
Uniqueness holds up to adding a constant (that is, up to translation
of the source). This is
what gives the compact-approximation potentials phi_R in Part 5, step
1. It is the modern form of Cordero-Erausquin and Klartag's moment
measures, and the "Fathi Stein kernel" viewpoint used in Part 6 goes
back to Fathi.
A2. Klartag Hessian estimate. If rho = e^{-V} on a smooth bounded
convex B with D^2 V >= eps I and V, together with its derivatives,
bounded on B, then the potential of A1 satisfies
0 < D^2 phi <= (4/eps) I .
The constant 4/eps is uniform in the set B and in the normalization.
This is the uniform upper curvature bound in Part 5, step 2, and the
source of every "tau is bounded above" statement.
A3. Borell's characterization. An even log-concave probability
measure on R^d is the pushforward of a log-concave density under an
affine map; if its support spans R^d, the map is invertible, so the
measure has a log-concave density. Used in the measure corollary
(Part 2) to replace a general even log-concave Radon measure on its
support subspace by a log-concave density.
A4. Saroglou's transfer theorem. The logarithmic Brunn-Minkowski
inequality for Lebesgue measure in dimension d implies the same
inequality in dimension d for every even log-concave density. Used in
the measure corollary after reducing to the support subspace of
dimension d.
A5. Elliptic regularity.
- Interior elliptic regularity: a distributional solution of an
elliptic equation with smooth coefficients is smooth. It gives
"g is smooth" in the density lemma (Part 7) from P g = 0, since
the principal symbol (xi.tau xi)^2 is positive.
- Evans-Krylov (nonhomogeneous form, Wang; book draft
Fernandez-Real and Ros-Oton): a uniformly elliptic concave
equation F(D^2 u) = f with measurable f bounded gives C^{2,alpha}
bounds. Applied to the spectral function tr s(D^2 phi_R), which
equals log det D^2 phi_R on the relevant set of Hessians.
- Schauder estimates then bootstrap to C^{k,alpha} for all k.
These give the compactness in Part 5, step 5, that produces the
limit phi.
A6. Standard real analysis.
- Dominated and monotone convergence: the q -> infinity and eps -> 0
limits in Part 4.
- Continuity of Lebesgue measure from above for decreasing
intersections of bounded sets: the dense-direction limit in Part 4.
- Weighted integration by parts in L^2(mu): the adjunction form in
Part 6 and the cutoff estimates in Parts 7-9.
============================================================
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============================================================
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============================================================
END.
============================================================
This document proves the even logarithmic Brunn-Minkowski conjecture
in every dimension, together with the symmetric L_p inequality and the
(B)-conjecture for even log-concave measures. The proof is the three
steps of Part 3: reduce volume to a variance inequality (Part 4),
build moment coordinates (Part 5), and prove the variance inequality
through the operator A (Parts 6-9). Every display is either elementary
or a named imported tool from the appendix. The single idea to carry
away is this: in the right coordinates the relevant diffusion has
nonnegative Hessian energy as an explicit sum of squares, and that is
exactly the curvature needed for log-concavity of volume.