==== TABLE OF CONTENTS ====
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//diamond.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//diamonds.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//forest-definition.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//forest-gates.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//forest.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//general-forests.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//introduction.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//main.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//obstruction.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired-cut.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//preamble.tex
Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//references.bib
===========================
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//diamond.tex ====
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.85,every node/.style={font=\small}]
\draw[thick] (0,0)--(2,0);
\fill (0,0) circle (2pt) node[left] {$u$};
\fill (2,0) circle (2pt) node[right] {$v$};
\draw[-{Stealth},thick] (2.8,0)--(4,0);
\coordinate (u) at (5,0); \coordinate (v) at (9,0);
\foreach \h/\lab in {1.4/1,0.45/2,-1.4/j}{
\draw[thick] (u)--(7,\h)--(v);
\fill (7,\h) circle (2pt) node[above] {$w_{\lab}$};
}
\node at (7,-0.5) {$\vdots$};
\fill (u) circle (2pt) node[left] {$u$};
\fill (v) circle (2pt) node[right] {$v$};
\node at (1,-1.9) {one old edge};
\node at (7,-1.9) {countably many two-edge paths};
\end{tikzpicture}
\caption{One diamond replacement, with only three of the countably many
paths shown. All new edges have length one. The old endpoints are at
distance two, as are any two distinct new middle vertices.}
\label{fig:diamond}
\end{figure}
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//diamond.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//diamonds.tex ====
\section{From approximate midpoints to diamonds}\label{sec:diamonds}
\subsection{An infinite family of approximate midpoints}
Support restrictions disappear when infinitely many separated
perturbations are available. This is the only compactness step in the
metric proof. Approximate-midpoint lenses are central to AMUC
\cite[Theorem~2.1]{DKRRZ2016}; the broader midpoint method is attributed
to Enflo's unpublished argument in \cite[Section~4]{BaudierEtAl2017}.
The configuration below is a separated midpoint family, called a spider
in \cite[Definition~3.7; Definition~6.15 and Proposition~6.16]{BassetLancienProchazka2026}.
We derive its precise squared deficit from Lemma~\ref{lem:paired}.
\begin{lemma}\label{lem:lens}
Let $R>0$, $\varepsilon>0$, $x\in X$, and $(y_j)_{j\ge1}\subset X$ satisfy
\[
\|x\pm y_j\|\le R,\qquad\|y_i-y_j\|\ge\varepsilon>0
\quad(i\ne j).
\]
Then
\begin{equation}\label{eq:lens}
R^2-\|x\|^2\ge\varepsilon^2/48.
\end{equation}
\end{lemma}
\begin{proof}
Convexity gives $\|x\|,\|y_j\|\le R$. Given $\eta>0$, choose
finite ancestral $H$ with $\|Q_Hx\|<\eta$. The bounded sequence
$(P_Hy_j)$ is in a finite-dimensional space, so some distinct $i,j$
satisfy $\|P_H(y_i-y_j)\|<2\eta$. Put $z=(y_i-y_j)/2$.
Since
\[
x+z=\tfrac12((x+y_i)+(x-y_j)),\quad
x-z=\tfrac12((x-y_i)+(x+y_j)),
\]
we have $\|x\pm z\|\le R$. Also
\[
\|P_Hz\|<\eta,\quad\|Q_Hz\|\ge\varepsilon/2-\eta,
\quad\|P_Hx\|\ge\|x\|-\eta.
\]
Both $P_Hx\pm Q_Hz$ have norm at most $R+2\eta$. By Lemma~\ref{lem:paired},
\[
(R+2\eta)^2\ge\|P_Hx\|^2+\|Q_Hz\|^2/12.
\]
For $0<\eta<\varepsilon/2$, let $\eta\downarrow0$ to obtain \eqref{eq:lens}.
\end{proof}
\subsection{Adding the loss over the diamond levels}
Figure~\ref{fig:diamond} shows the replacement whose loss we now estimate.
\input{figures/diamond}
Each parent edge will consume the same positive amount of squared
stretch. Earlier quantitative diamond arguments accumulate convexity
losses across levels \cite[Lemma~2.2]{LeeNaor2004}, or use a
self-improvement induction \cite[Lemma~1 and Proposition~2]{JohnsonSchechtman2009}.
The countably branching AMUC form appears in
\cite[Section~4]{BaudierEtAl2017}. Here Lemma~\ref{lem:lens} supplies
a fixed squared loss, which we can add directly.
\begin{proof}[Proof of the distortion assertion in Theorem~\ref{thm:main}]
Let $f:D_k\to X$ satisfy $s d_k\le\|f(u)-f(v)\|\le Ds d_k$.
Divide by $s$, so $f$ is noncontracting and $D$-Lipschitz. Distances
between old vertices double at each refinement: a new path between
old vertices consists of two-edge passages which either return to
their starting vertex or contract to an old edge, and every old path
can be lifted with twice its length. Therefore the endpoints of an
edge of $D_m$ have distance $2^{k-m}$ in $D_k$. Distinct middle
vertices replacing that edge also have distance $2^{k-m}$ in $D_k$.
Put
\[
L_m=\sup_{\{u,v\}\in E(D_m)}
\frac{\|f(v)-f(u)\|}{2^{k-m}},\qquad 0\le m\le k.
\]
These suprema need not be attained; all that is used is
$1\le L_m\le D$. Fix one parent edge at level $m0).
\end{equation}
This also explains why reflexivity is used for the example's structural
identification, rather than as a hypothesis of the paired estimate.
For $T=\mathbb N^{<\omega}$, the coordinate completion $X(T)$ is
the countably branching James-tree predual $B_\infty$, as described
in Banakh~\cite[\S2.II]{Banakh2000} following
Ghoussoub--Maurey--Schachermayer. We give its path obstruction in
the displayed norm. Each root-path
functional $p_v=\sum_{w\preceq v}e_w^*$ again has norm one. The
antichain calculation in \eqref{eq:antichain-isometry} uses only
disjoint segments and applies to $X(T)$ as well. Thus, for any
$\phi\in X(T)^*$ and finitely many children of a fixed vertex,
\[
\left|\sum_j a_j\phi(e_{v^\frown j}^*)\right|
\le\|\phi\|\left(\sum_j a_j^2\right)^{1/2}.
\]
Taking $a_j=\phi(e_{v^\frown j}^*)$ shows that these evaluations are
square summable, and hence tend to zero. The child increments are
therefore weakly null against the whole continuous dual. Applying
Lemma~\ref{lem:bounded-tree} with $m=M=c=1$ proves
$\overline\delta_N(\alpha/(2\beta))=0$ for every equivalent norm
$\alpha\|\cdot\|\le N\le\beta\|\cdot\|$ on this coordinate
completion. This argument does not require it to be reflexive.
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//general-forests.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//introduction.tex ====
\section{Introduction}\label{sec:introduction}
Asymptotic convexity describes the convexity that remains after finitely
many directions are discarded. The question considered here is whether
symmetric midpoint convexity can persist in a reflexive Banach space
which admits no asymptotically uniformly convex equivalent norm.
All spaces and scalars are real. For an infinite-dimensional Banach
space $Y$ with norm $N$, write $\operatorname{cof}(Y)$ for the closed
finite-codimensional subspaces of $Y$. For $t>0$, define
\begin{align}
\overline\delta_N(t)
&=\inf_{N(x)=1}\sup_{F\in\operatorname{cof}(Y)}
\inf_{y\in F,\,N(y)=1}\bigl(N(x+ty)-1\bigr),
\label{eq:auc-modulus}\\
\widehat\delta_N(t)
&=\inf_{N(x)=1}\sup_{F\in\operatorname{cof}(Y)}
\inf_{y\in F,\,N(y)=1}
\left(\frac{N(x+ty)+N(x-ty)}2-1\right).
\label{eq:average-modulus}
\end{align}
The norm is \emph{asymptotically uniformly convex} (AUC) if
$\overline\delta_N(t)>0$ for all $t>0$, and \emph{asymptotically
midpoint uniformly convex} (AMUC) if $\widehat\delta_N(t)>0$ for all
$t>0$. Thus AUC requires a gain in every sufficiently far-out direction;
AMUC permits the two opposite directions to be tested together. A norm
$N$ is equivalent to a given norm $\|\cdot\|$ if
$a\|x\|\le N(x)\le b\|x\|$ for some $00$ satisfies
\begin{equation}\label{eq:embedding-definition}
s\,d_k(u,v)\le\|f(u)-f(v)\|\le Ds\,d_k(u,v)
\qquad(u,v\in D_k).
\end{equation}
The family embeds uniformly when one finite $D$ works for all $k$.
For finite binary diamonds, Lee and Naor obtained quantitative $L_p$
distortion bounds by summing convexity losses \cite[Lemma~2.2]{LeeNaor2004},
and Johnson and Schechtman developed a self-improvement proof
\cite[Section~4]{JohnsonSchechtman2009}.
Baudier, Causey, Dilworth, Kutzarova, Randrianarivony, Schlumprecht,
and Zhang proved that an equivalent AMUC norm prevents uniform
embeddings of the countably branching diamonds
\cite[Theorem~4.1]{BaudierEtAl2017}. For reflexive spaces
with unconditional asymptotic structure, they also proved that failure
of AUC renormability implies uniform diamond embeddings
\cite[Theorem~B and Corollary~5.3]{BaudierEtAl2017}. Swift extended the embedding construction to countably branching bundle
graphs under the same reflexive, unconditional, and non-AUC-renormable
hypotheses \cite[Theorem~3.2 and Corollary~3.4]{Swift2018}. Perreau
subsequently treated duals of separable spaces with weak-star unconditional asymptotic
structure, retaining unconditional asymptotic structure in the reflexive
formulation \cite[Theorems~4.8 and~5.3]{Perreau2021}. The example below
shows that the diamond implication fails for general reflexive spaces.
More recently, Basset, Lancien, and Proch\'azka related AMUC to separated
families of midpoints and obtained the diamond obstruction even when
the maps are controlled only on specified vertex pairs
\cite[Definition~6.15 and Propositions~6.16 and~6.20]{BassetLancienProchazka2026}.
Malthaner obtained quantitative obstructions for countably branching
diamonds using a Matou\v{s}ek-type coloring and path argument
\cite[Theorems~3.5 and~3.7]{Malthaner2025}. We use the full
bi-Lipschitz convention \eqref{eq:embedding-definition} and prove the
required midpoint estimate directly.
\subsection{The example and result}
\input{sections/forest-definition}
For the classical binary James tree space, Girardi proved AUC of both
the coordinate predual and the full dual
\cite[Theorems~3 and~5]{Girardi2001}. The countable branching and finite,
unbounded component heights used here lead to the following different
behavior.
\begin{theorem}\label{thm:main}
The space $X=J^*$ defined by \eqref{eq:segment-norm} is infinite
dimensional, separable, and reflexive, and has no equivalent AUC norm.
Its given norm satisfies
\[
\widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).
\]
For every integer $k\ge0$, every embedding of $D_k$ into $X$ has
distortion at least $\sqrt{1+k/12}$.
\end{theorem}
This gives a negative answer to the reflexive diamond question recorded
as Problem~39 in \cite[author manuscript p.~388]{BaudierLancien2025}. The metric obstruction is
unchanged by equivalent renorming, although the numerical constants
refer to the specified dual norm.
\subsection{How the argument fits together}
Section~\ref{sec:forest} establishes reflexivity and the coordinate
projections used throughout the proof. A sum of coordinate functionals
along one root path has norm one, while its next coordinate increment
is a weakly null unit vector as the child varies. In
Section~\ref{sec:no-auc}, a hypothetical AUC norm forces repeated
multiplicative gains along one such path. Arbitrarily high finite trees
contradict its bounded norm.
The midpoint argument in Section~\ref{sec:paired} must control segment
tests crossing from a finite ancestral set (one closed under taking ancestors) to its
complement. Simply
adding norming tests for the two parts creates uncontrolled positive
cross terms. We add one correction at their first exit vertices. The
correction makes both resulting tests inexpensive, and its evaluations
cancel when the two endpoint norms are averaged. Lemma~\ref{lem:paired}
expresses the result as a lower bound in terms of the squared head and
tail norms.
Section~\ref{sec:general-forest} extends this estimate to coordinate
completions on general rooted forests.
Finally, the infinitely many middle vertices of a diamond supply a
separated family of approximate midpoints. Two of their finite
coordinate projections can be made arbitrarily close. Their
half-difference therefore reduces the metric problem to the head--tail
estimate. Section~\ref{sec:diamonds} turns this into a loss in squared
stretch at every replacement and sums the losses. This gives a direct
quantitative instance of the midpoint and self-improvement methods for
these graphs \cite[Section~4]{BaudierEtAl2017}.
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//introduction.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//main.tex ====
\input{preamble}
\title{Asymptotic midpoint uniform convexity\texorpdfstring{\\}{ }and unbounded diamond distortion\texorpdfstring{\\}{ }in a reflexive tree space}
\hypersetup{pdftitle={Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space}}
\begin{document}
\maketitle
\begin{abstract}
We construct a separable reflexive real Banach space whose given norm
is asymptotically midpoint uniformly convex but which admits no
asymptotically uniformly convex equivalent norm. Its averaged midpoint
modulus is at least $\sqrt{1+t^2/12}-1$, and the countably branching
diamond of depth $k$ has distortion at least $\sqrt{1+k/12}$ in this
space. This gives a negative answer to the reflexive diamond converse
for asymptotic uniform convexifiability.
\end{abstract}
\input{sections/introduction}
\input{sections/forest}
\input{sections/obstruction}
\input{sections/paired}
\input{sections/general-forests}
\input{sections/diamonds}
\par
\begingroup
\raggedright
\bibliographystyle{plainnat}
\bibliography{references}
\endgroup
\end{document}
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//main.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//obstruction.tex ====
\section{Why no equivalent norm is AUC}\label{sec:no-auc}
The renorming obstruction uses bounded root paths and weakly null
child increments. We first isolate the elementary finite-codimensional
argument; its midpoint counterpart will be proved separately.
\begin{lemma}[Modulus conventions]\label{lem:modulus-conventions}
For an infinite-dimensional real Banach space $Y$ with norm $N$, let
$m_N$ be the modulus obtained from \eqref{eq:average-modulus} by replacing
the average by $\max\{N(x+ty),N(x-ty)\}$. Then, for $t>0$,
\[
0\le\overline\delta_N(t)\le\widehat\delta_N(t)
\le m_N(t)\le2\widehat\delta_N(t).
\]
For either symmetric modulus the inner infimum is unchanged if
$N(y)=1$ is replaced by $N(y)\ge1$. Both symmetric moduli, including
their centerwise versions before taking the infimum over $x$, are
nondecreasing. Positivity on $(0,1)$ therefore implies positivity at
every positive radius.
\end{lemma}
\begin{proof}
For fixed $x,y$, the average and maximum of the two signed norms are
even convex functions of the radius, hence nondecreasing on $[0,\infty)$.
This proves the assertions about radii and monotonicity. A norming
functional for a unit center $x$ makes both signed norms at least one
on its kernel. Intersecting any chosen finite-codimensional subspace
with this kernel proves nonnegativity of the one-sided modulus and
that the maximum excess is at most twice the average excess, after
taking the supremum over subspaces. The average is pointwise at most
the maximum. Finally, on a fixed subspace the infimum of the one-sided
excess is a lower bound for each of the two signed excesses, since
$y$ and $-y$ both belong to that subspace. Taking the successive
suprema and infima proves the remaining comparison.
\end{proof}
\begin{lemma}[Bounded trees]\label{lem:bounded-tree}
Let $Y$ be a real Banach space. Suppose that for every integer $h\ge1$
there are vectors $(u_s^{(h)})_{s\in\mathbb N^{\le h}}$ and constants
$m,M,c>0$, independent of $h$, such that
\[
m\le\|u_s^{(h)}\|\le M,
\qquad
\|u_{s^\frown j}^{(h)}-u_s^{(h)}\|\ge c\quad(|s|\beta M$ and make this
choice successively along a branch of that tree. The resulting norm
exceeds $\beta M$, a contradiction. The increments in particular
force $Y$ to be infinite dimensional, and Lemma~\ref{lem:modulus-conventions}
gives the reverse nonnegative inequality in \eqref{eq:bounded-tree-scale}.
An infinite tree may be restricted to any sufficiently large height.
\end{proof}
We apply the lemma to the forest. For a node $v$ in the component with
root $r$, put $p_v=\sum_{r\preceq w\preceq v}e_w^*$. The root path
is one segment test, so $\|p_v\|\le1$; evaluation at a coordinate
on that path gives equality. At a nonterminal node the child increments
are $e_{v^\frown j}^*$, each of norm one. They tend weakly to zero:
$X^*=J$, and every $u\in J$ has square-summable coordinates, so
$u(v^\frown j)\to0$. We therefore have $m=M=c=1$ in
Lemma~\ref{lem:bounded-tree}. For every equivalent norm
$\alpha\|\cdot\|\le N\le\beta\|\cdot\|$,
\[
\overline\delta_N(\alpha/(2\beta))=0.
\]
This proves the renorming assertion in Theorem~\ref{thm:main}.
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//obstruction.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired-cut.tex ====
\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
\fill[blue!5] (-.4,-.6) rectangle (6.25,1.25);
\fill[orange!8] (6.25,-.6) rectangle (9.6,1.25);
\draw[thick] (0,0)--(9.2,0);
\draw[dashed,gray] (6.25,-.7)--(6.25,.8);
\foreach \x in {0,1,2,3,4,5,6,6.6,7.5,8.3,9.2}
\fill (\x,0) circle (1.7pt);
\node[below] at (0,0) {$a$};
\node[below] at (4,0) {$a'$};
\node[below] at (6,0) {$v$};
\node[below] at (6.6,0) {$g$};
\node[below] at (9.2,0) {$w$};
\draw[decorate,decoration={brace,amplitude=4pt}] (0,.35)--(6,.35)
node[midway,above=5pt] {head sum $\alpha$};
\draw[decorate,decoration={brace,amplitude=4pt}] (4,.95)--(6,.95)
node[midway,above=5pt] {selected suffix $\alpha'$};
\draw[decorate,decoration={brace,amplitude=4pt}] (6.6,.35)--(9.2,.35)
node[midway,above=5pt] {tail sum $\beta$};
\node at (2,-.5) {$H$};
\node at (8,-.5) {$\mathcal F\setminus H$};
\end{tikzpicture}
\caption{The threshold cut for a crossing segment. Its head has
$f$-sum $\alpha$ and its selected suffix has $f$-sum $\alpha'$.
When $\alpha\beta>0$ and $|\alpha|>2sm_g$, the sign choice gives
$\alpha\alpha'<0$ and $|\alpha'|\ge2sm_g$.
Cutting immediately before $a'$ therefore increases the squared
$f$-test by at least $2s\alpha\beta$. The vertex spacing is schematic.}
\label{fig:paired-cut}
\end{figure}
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired-cut.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired.tex ====
\section{A paired extension of segment tests}\label{sec:paired}
The renorming obstruction is proved. To establish midpoint convexity,
we now show that a symmetric pair must grow when its center and
displacement occupy a finite ancestral set and its complement. The
squared-norm form will also be the input for the diamond argument.
\begin{lemma}\label{lem:paired}
Let $H\subset\mathcal F$ be finite and ancestral. If $x,y\in X$
satisfy $x=P_Hx$ and $y=Q_Hy$, then
\begin{equation}\label{eq:paired}
\frac{\|x+y\|+\|x-y\|}{2}
\ge\sqrt{\|x\|^2+\|y\|^2/12}.
\end{equation}
More generally, for all $s,t>0$,
\begin{equation}\label{eq:paired-two-parameters}
\frac{\|x+ty\|+\|x-ty\|}{2}
\ge\frac{\|x\|+st\|y\|}{\sqrt{1+12s^2}}.
\end{equation}
\end{lemma}
\begin{proof}
We first prove, for every $s>0$,
\begin{equation}\label{eq:paired-parameter}
\frac{\|x+y\|+\|x-y\|}{2}
\ge\frac{\|x\|+s\|y\|}{\sqrt{1+12s^2}}.
\end{equation}
Suppose initially that $y$ has finite support. Choose finite tests
$f,k\in J$, supported in $H$ and its complement respectively, with
$\rho(f),\rho(k)\le1$ and pairings as close as desired to $\|x\|$ and
$\|y\|$. The naive tests $f+sk$ and $f-sk$ can have large positive crossing
terms. We will add the same vector to both tail tests; its contribution
to the endpoint average will cancel. To control its size, at each gate
$g$ put
\[
m_g=\sup_{w\succeq g}|k([g,w])|.
\]
Only finitely many $m_g$ are nonzero, and $\sum_gm_g^2\le1$:
choose an almost maximizing initial segment in each of finitely many
distinct cones and use their disjointness.
For a gate with parent $v\in H$, consider the suffixes $[a,v]$ whose
$f$-sum has magnitude at least $2sm_g$. If $m_g>0$ and such suffixes
exist, choose the shortest one and let $\sigma_g$ be opposite to the
sign of its sum. In all other cases put $\sigma_g=1$. Set
\[
c=\sum_g\sigma_gm_ge_g,\qquad v^+=c+k,\qquad v^-=c-k.
\]
The gates form an antichain, so $\rho(c)\le1$ and $\rho(v^\pm)\le2$.
Every initial segment in the cone of $g$ has $v^\pm$-sum of sign
$\sigma_g$ or zero, and magnitude at most $2m_g$.
We claim $\rho(f+sv^\pm)^2\le1+12s^2$. Fix a disjoint family of
segments and denote their $f$- and $v^\pm$-sums by $\alpha_i$ and
$\beta_i$. We have $\sum_i\beta_i^2\le4$. A positive cross term
$\alpha_i\beta_i$ can occur only at a crossing segment, with some
gate $g$; different crossing segments use different gates. If
$|\alpha_i|\le2sm_g$, then $2s\alpha_i\beta_i\le8s^2m_g^2$.
Otherwise the head itself reaches the threshold, so it contains the
shortest selected suffix. Write $\alpha'_i$ for that suffix's $f$-sum.
Since $\alpha_i\beta_i>0$ and $\beta_i$ has sign $\sigma_g$, the choice
of $\sigma_g$ gives $\alpha_i\alpha'_i<0$. Thus the selected suffix is
proper, and $|\alpha'_i|\ge2sm_g$ by its selection.
Splitting it off the head increases the sum of squared $f$-sums by
\[
(\alpha_i-\alpha'_i)^2+(\alpha'_i)^2-\alpha_i^2
=2(\alpha'_i)^2+2|\alpha_i\alpha'_i|
\ge4s|\alpha_i|m_g\ge2s\alpha_i\beta_i.
\]
Figure~\ref{fig:paired-cut} shows the proper suffix and the cut,
with the segment index suppressed.
\input{figures/paired-cut}
These splits involve distinct original segments, so all can be made
simultaneously. Their pieces and the remaining heads are still
disjoint. Thus $\sum_i\alpha_i^2$ together with all the large
positive cross terms is at most $\rho(f)^2\le1$. The small cross terms
cost at most $8s^2$, and the pure tail terms cost at most $4s^2$.
We have controlled both corrected tests. It remains to use them
together so that their common correction disappears.
Test $x+y$ against $(f+sv^+)/\sqrt{1+12s^2}$ and $x-y$ against
$(f+sv^-)/\sqrt{1+12s^2}$. The mixed support terms vanish, and the
two evaluations involving $c$ cancel in the average. Letting the
norming errors tend to zero proves \eqref{eq:paired-parameter}. Finite tails are dense in
$Q_HX$, so \eqref{eq:paired-parameter} holds for every $y=Q_Hy$. If $\|x\|>0$, maximize
its right side by taking $s=\|y\|/(12\|x\|)$; zero values follow
by a limit. The maximum is the right side of \eqref{eq:paired}. Replacing $y$ by $ty$ in \eqref{eq:paired-parameter} proves
\eqref{eq:paired-two-parameters}.
\end{proof}
\begin{proof}[Proof of the midpoint assertion in Theorem~\ref{thm:main}]
For a unit $x\in X$, choose $H$ with $\|x-P_Hx\|<\eta$. For
every unit $y\in\ker P_H$, Lemma~\ref{lem:paired} applied to $P_Hx$ and $ty$ gives
\[
\frac{\|x+ty\|+\|x-ty\|}{2}
\ge\sqrt{(1-\eta)^2+t^2/12}-\eta
\qquad(0<\eta<1).
\]
Take the infimum over these $y$, the supremum over finite-codimensional
subspaces, and let $\eta\downarrow0$. This proves
the asserted bound, since it is independent of the unit center $x$.
\end{proof}
==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//paired.tex ====
==== BEGIN Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//preamble.tex ====
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==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//preamble.tex ====
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==== END Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space//references.bib ====