we-solve (beta)

Math-2695 · mathematics · Analysis

Asymptotic midpoint renormability versus AUC

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every infinite-dimensional real Banach space that admits an asymptotically midpoint uniformly convex norm also admit an equivalent asymptotically uniformly convex norm? Positivity of the midpoint modulus is in the finite-codimensional averaged two-sided convention; AUC uses the corresponding one-sided convention.

Short English statement

Must every infinite-dimensional real Banach space that admits an asymptotically midpoint uniformly convex norm also admit an equivalent asymptotically uniformly convex norm? Positivity of the midpoint modulus is in the finite-codimensional averaged two-sided convention; AUC uses the corresponding one-sided convention.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2695-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2695

The question asks whether midpoint asymptotic uniform convexity forces an equivalent asymptotically uniformly convex norm. The reflexive tree-norm manuscript already claims a space satisfying the midpoint property but having no AUC renorming. This is a counterexample to the complete implication. The other selected manuscripts give further examples with their stated tree or $L^1$ constructions. The first manuscript also supports the full negative answer; no undecided clause is supplied only by a weaker companion.

Asymptotic midpoint renormability versus AUC

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 331; 2026-09-27), the asserted result is that a separable reflexive tree-norm space is asymptotically midpoint uniformly convex with modulus at least $\sqrt{1+t^2/12}-1$ but has no AUC renorming, and depth-$k$ diamonds incur distortion at least $\sqrt{1+k/12}$; In manuscript 2 (family 331; 2026-09-27), the asserted result is that a real measure-precompact Daugavet subspace of $L^1$ has exact averaged midpoint modulus $\max\{t/2,t-1\}$ and one-sided modulus $\max\{0,t-2\}$, and no equivalent AUC norm; In manuscript 3 (family 331; 2026-09-27), the asserted result is that path products of independent exponential or Gaussian-square variables span real $L^1$ spaces with positive midpoint moduli but no AUC renorming; In manuscript 4 (family 331; 2026-09-27), the asserted result is that four bounded-tree-potential norms have averaged midpoint lower bound $\sqrt{1+t^2/4}-1$ for $0<t<1$ but no AUC renorming, with the stated reflexive Hilbert-sum variants; In manuscript 5 (family 331; 2026-09-27), the asserted result is that recursive two-field tree norms have cubic or sixth-power midpoint bounds, including reflexive examples with no AUC equivalent norm; In manuscript 6 (family 331; 2026-09-27), the asserted result is that segment-forest duals and the infinite-height coordinate predual have small symmetric-lens tails and midpoint convexity, despite failing every AUC renorming, with reflexivity for the stated finite-height forest dual.

Mathematical statement of the claimed result

Manuscript 1.
Let $\mathcal F=\coprod_{h\ge1}\{h\}\times\mathbb N^{\le h}$ be a forest of countably branching finite-height trees. Complete finitely supported real functions under \[ \rho(u)=\sup_{S_1,\ldots,S_m}\left(\sum_{i=1}^m(\sum_{v\in S_i}u(v))^2\right)^{1/2}, \] where the supremum ranges over finite families of pairwise disjoint ancestral segments, and call the completion $J$. The claim states that $X=J^*$ is separable, infinite dimensional and reflexive, admits no equivalent asymptotically uniformly convex norm, and its dual norm satisfies \[ \widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\quad(t>0). \] Here $\widehat\delta_X(t)=\inf_{\|x\|=1}\sup_F\inf_{y\in F,\|y\|=1}((\|x+ty\|+\|x-ty\|)/2-1)$, with $F$ closed and finite-codimensional. Every embedding of the $k$th countably branching diamond graph, with shortest-path metric, has distortion at least $\sqrt{1+k/12}$ for every $k\ge0$.
Manuscript 2.
Take a probability space $(\Omega,\mathbb P)$ and a real closed subspace $E\subseteq L^1(\Omega,\mathbb P)$ with infinite dimension. Require total boundedness of its unit ball for the topology of convergence in measure. For the restricted $L^1$ norm also require the Daugavet equation $\|I+T\|=1+\|T\|$ for every rank-one operator $T$. For $\|x\|=1$ define \[ H_E(x,t)=\sup_F\inf_{y\in F,\|y\|=1}((\|x+ty\|+\|x-ty\|)/2-1), \quad D_E(x,t)=\sup_F\inf_{y\in F,\|y\|=1}(\|x+ty\|-1), \] with $F$ closed finite-codimensional. The claimed exact identities are \[ H_E(x,t)=\max\{t/2,t-1\},\qquad D_E(x,t)=\max\{0,t-2\}\quad(t>0). \] These also give the infima over unit $x$, namely the averaged midpoint and AUC moduli. Such spaces are asserted to exist by the stated Kadets--Werner construction and admit no equivalent AUC norm.
Manuscript 3.
On the tree $\mathbb N^{<\omega}$ take independent copies $W_v$ of a positive nonconstant real variable with $\mathbb EW=1$, $\mathbb EW^2<\infty$. Put $P_{\varnothing}=1$, $P_v=P_{v^-}W_v$ and $X_W=\overline{\operatorname{span}_{\mathbb R}\{P_v\}}^{L^1}$. Every such space is claimed infinite dimensional and to admit no equivalent AUC norm. With inherited norm, let $\widehat\delta$ use the averaged signed norms, infimum over unit centers, supremum over closed finite-codimensional $F$ and infimum over $y\in F$ of norm at least one. For exponential $W$ and $W=G^2$, $G$ standard normal, define $c(K)=\mathbb E(|G|-2K)_+/16$. The claimed positive estimates are \[ \widehat\delta_{X_{\rm exp}}(t)\ge(7t/30)c(10/t),\quad \widehat\delta_{X_{\rm sq}}(t)\ge\Pr(|G|\ge4\sqrt2(2+\sqrt2)/t)/8, \quad\widehat\delta_{X_{\rm sq}}(t)\ge\Pr(|G|\ge8\sqrt6/t)/4 \] for every $t>0$.
Manuscript 4.
Let $T=\mathbb N^{<\omega}$, root $o$, and $P_f(s)=\sum_{u\in I,\ u\preceq s}f_u$. Four symmetric test sets all require $|P_f(s)|\le1$. The root-free sibling set uses $I=T\setminus\{o\}$ and $\sum_{u\in\operatorname{ch}(r)}f_u^2\le1$; the root-inclusive sibling set uses $I=T$ with that same constraint. The antichain set uses finitely supported $f$ on $T$ and $\sum_{u\in A}f_u^2\le1$ for every finite antichain. The global set uses $f\in\ell_2(T)$ with $\sum_Tf_u^2\le1$. Complete finite vectors in $\|x\|_K=\sup_{f\in K}\sum_s f_sx_s$ to obtain $X_K$. Each is claimed separable infinite dimensional, has no equivalent AUC norm, and has averaged midpoint modulus \[ \widehat\delta_{X_K}(t)\ge\sqrt{1+t^2/4}-1\quad(0<t<1). \] For the segment construction, use disjoint component trees $\mathcal T_n=\mathbb N^{\le n}$, $n\ge1$. On nonnegative arrays recursively define \[ q_t(m)=m_t+(\sum_{u\in\operatorname{ch}(t)}q_u(m)^2)^{1/2},\qquad M_n(m)=q_{\varnothing}(m). \] A segment array $(a_{tv})_{t\preceq v}$ has start masses $m_t(a)=\sum_{v\succeq t}|a_{tv}|$ and output $(Pa)_s=\sum_{t\preceq s}\sum_{v\succeq s}a_{tv}$. Set \[ \|x\|_{S,n}=\inf_{Pa=x}M_n(m(a)),\qquad X_S=\left(\bigoplus_{n\ge1}(\ell_2(\mathcal T_n),\|\cdot\|_{S,n})\right)_{\ell_2}. \] The claimed complete component norms satisfy $\|x\|_2/(n+1)\le\|x\|_{S,n}\le\sqrt{n+1}\|x\|_2$, and supply a reflexive $X_S$ with contractive finite ancestral coordinate projections, unit coordinate and root-to-node path vectors, and weakly null sibling sequences; its maximum midpoint modulus satisfies \[ \delta^{\max}_{X_S}(t)\ge\sqrt{1+t^2/64}-1\quad(t>0). \] Both midpoint moduli take the infimum over unit centers, supremum over closed finite-codimensional subspaces and infimum over their large directions; the first uses the average of the two signed norms, the second their maximum.
Manuscript 5.
For finitely supported real vectors on a rooted countably branching tree, compute upward from zero fields below the support: \[ A_s=(\sum_{c\in\operatorname{ch}(s)}P_c^2)^{1/2},\quad B_s=(\sum_{c\in\operatorname{ch}(s)}Q_c^2)^{1/2},\quad P_s=\max\{A_s,v_s+B_s\},\quad Q_s=\max\{B_s,-v_s+A_s\}. \] On $\mathbb N^{<\omega}$ let $N_{\Sigma}=P_o+Q_o$. Join disjoint trees $\mathbb N^{\le n}$, $n\ge1$, under a new root $\rho$; with $v_{\rho}=0$ let $N_0=P_{\rho}=Q_{\rho}$, and with that coordinate free let $N_J=P_{\rho}+Q_{\rho}$. Complete to $X_{\Sigma},X_0,X_J$. All are claimed Banach spaces with no equivalent AUC norm, and $X_0,X_J$ are reflexive. The averaged midpoint estimates are \[ \widehat\delta_{N_{\Sigma}}(t),\widehat\delta_{N_J}(t)\ge t^3/128\quad(0<t<1). \] If $H$ is finite ancestral, $x$ supported in $H$, and $0\ne z$ vanishes on $H$, the further estimate is \[ \frac{N_0(x+z)+N_0(x-z)}2\ge N_0(x)+\frac{N_0(z)^3}{8(2N_0(x)+N_0(z))^2}. \] Coordinate restrictions to finite ancestral sets are contractive, their kernels finite-codimensional, and finite-support vectors dense there.
Manuscript 6.
Complete finitely supported real arrays on the forest $\coprod_{h\ge1}\mathbb N^{\le h}$ in the norm $\rho(a)=\sup(\sum_j|\sum_{v\in S_j}a_v|^2)^{1/2}$ over finite disjoint ancestral segments, and set $X=J^*$. Define the same segment completion $J_{\infty}$ on $\mathbb N^{<\omega}$ and let $B_{\infty}$ be the closed span of coordinate evaluations in $J_{\infty}^*$. For either $Y=X$ or $B_{\infty}$, finite ancestral $H$, $P_H$ coordinate restriction, $Q_H=I-P_H$, $x=P_Hx$ and $\|x\pm y\|\le R$, the asserted lens estimate is \[ \|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}. \] Its averaged midpoint consequence is $\widehat\delta_Y(t)\ge(\sqrt{1+t^2/4}-1)/2$ for every $t>0$. The finite-height dual is claimed separable reflexive, with dense coordinate span and contractive ancestral projections. For any unbounded sequence of finite component heights and equivalent norm $N$ satisfying $a\|x\|\le N(x)\le b\|x\|$, it claims \[ \overline\delta_N(a/b)=\overline\delta_N(a/(2b))=0, \] so there is no equivalent AUC norm. Here the one-sided modulus is $\inf_{N(x)=1}\sup_F\inf_{y\in F,N(y)=1}(N(x+ty)-1)$.

Outline of the argument

For manuscript 1 (family 331; 2026-09-27), paired segment tests bound approximate midpoint tails in the coordinate completion. Fresh layers along a branching diamond accumulate an energy loss at each level, while uniformly bounded branches persist under equivalent norms and obstruct one-sided asymptotic convexity. For manuscript 2 (family 331; 2026-09-27), a quantitative finite-dimensional enlargement constructs the required subspace. Compact clipping leaves a fixed mass outside each head, while almost diametral weak-neighborhood points give matching upper tests; the same directions exclude one-sided convexification under every renorming.

For manuscript 3 (family 331; 2026-09-27), fresh multipliers are conditioned along finite tree heads. Gaussian scale-mixture identities or scalar square estimates force uniform excess for every coefficient list, giving midpoint growth, whereas bounded positive-mean paths obstruct every one-sided renorming. For manuscript 4 (family 331; 2026-09-27), two inward tests split the tail of an arbitrary dual potential relative to a head norming functional. Clipping and subtree quotas bound path coefficients, while newly exposed coordinate energy gives midpoint growth; bounded branches survive all equivalent norms.

For manuscript 5 (family 331; 2026-09-27), least positive fields represent the vector through their difference. Root defects propagate until a stopping boundary, where accumulated Euclidean energy yields the midpoint estimate; finite-height Hilbert equivalence proves reflexivity, while bounded paths prevent AUC renorming. For manuscript 6 (family 331; 2026-09-27), finite convex laws are recovered by clipping cumulative segment sums and matching opposite masses at boundary crossings. Endpoint comparisons bound the tail by the squared-radius deficit, and completed atomic laws transfer the estimate to the specified spaces.

Formal verification

For manuscript 1 (family 331; 2026-09-27), the documented Lean scope is the specified segment-space dual, its midpoint modulus $\sqrt{1+t^2/12}-1$, and its separability, reflexivity, renorming obstruction, and diamond-distortion estimate; extensions to general forests concern a closed coordinate span and do not assert these conclusions for every full forest dual.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

For manuscript 2 (family 331; 2026-09-27), OpenAI lists Lean statements for this assertion: a real measure-precompact Daugavet subspace of $L^1$ has exact averaged midpoint modulus $\max\{t/2,t-1\}$ and one-sided modulus $\max\{0,t-2\}$, and no equivalent AUC norm.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 3 (family 331; 2026-09-27), OpenAI lists Lean statements for this assertion: path products of independent exponential or Gaussian-square variables span real $L^1$ spaces with positive midpoint moduli but no AUC renorming.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 4 (family 331; 2026-09-27), the documented Lean scope is the stated tree-potential completions and their midpoint modulus, completeness, separability, and obstruction to asymptotic uniform convexity; the selected path-cost and comparison statements retain their support and head-tail hypotheses, and the remaining auxiliary assertions are outside this scope.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 5 (family 331; 2026-09-27), the documented Lean scope is the root-sum and joined-space midpoint bound $t^3/128$ for $0<t<1$, the stated zero-root tail estimates with finite-ancestor support, and the selected reflexivity conclusions for the joined and zero-root spaces; variable-exponent comparisons retain their sequence and head conditions.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 6 (family 331; 2026-09-27), the documented Lean scope is the tail estimate $2\sqrt{R^2-\lVert x\rVert^2}$ for centers supported on a finite ancestral head, under the two radius inequalities, in the specified finite-height dual and infinite-tree coordinate settings; the selected duality and renorming conclusions retain those forest assumptions.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are BoundedTreePotentials.lean, DaugavetModuli.lean, DiamondDistortion.lean, ForestSpace.lean, IndependentProducts.lean, MidpointLenses.lean, RecursivePotentials.lean.

Their named declarations include OAI.BoundedTreePotentials.TreeCalculus.main_counterexample, OAI.ComparatorModel.RecursivePotentials.main, OAI.DiamondDistortion.headline_all_pairs, OAI.ExactModuli.exact_example, OAI.ExactModuli.exact_main, OAI.ExactModuli.kadets_werner_existence, OAI.ForestSpace.main_theorem, OAI.IndependentProducts.main_exponential, OAI.IndependentProducts.main_gaussian_12, OAI.IndependentProducts.main_gaussian_24, OAI.IndependentProducts.main_general, OAI.SegmentLenses.midpoint_lens_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →