Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.03204

A polynomial bound in Dvoretzky's theorem

Boaz Klartag·Shahar Moshe
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.

Explore related subjects

math.FAmath.MGmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Boaz Klartag, Shahar Moshe. 2026-10-02. A polynomial bound in Dvoretzky's theorem. https://arxiv.org/abs/2610.03204

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality

Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^αx\|_X \lesssim \|\partial x\|_Y$ for some $0 < α< 1$, combined with the condition $0 \in ρ(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in ρ(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.

math.FA↗
arXiv2026

Linear maps preserving families of C-symmetric operators

Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$. We characterize the bounded bijective complex-linear maps $T$ on $\BH$ for which there exists a bijection $ψ$ of the set of conjugations,preserving commutativity in both directions, such that $T$ maps the space of $C$-symmetric operators onto the space of $ψ(C)$-symmetric operators for every conjugation $C$. We show that this condition is equivalent to the existence of a bijection $φ$ of the set of all orthonormal bases such that $T$ maps the set of all $(e_n)$-diagonal operators onto the set of all $φ((e_n))$-diagonal operators. We also characterize the corresponding preservers in dimension two using Pauli coordinates, without assuming the commutativity condition on $ψ$.

math.FA↗
arXiv2026

Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces

Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0 k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(Ω), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.

math.FA↗