arXiv subjects

Qingjin Cheng

Publications and source records attributed to Qingjin Cheng.

6 recordsLinked to original sources

arXiv2026

On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson

Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $ψ:S_V\to S_{\ell_2}$ such that the restrictions $ψ|_{S_{V_j}}$, $j=1,2,\ldots$, are isomorphisms from $S_{V_j}$ onto $S_{H_j}$. The Kasparov-Yu problem asks whether the Banach space $c_0$ of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that $c_0$ admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum $μ_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\ell_\infty^n}$ to $S_{\ell_2^n}$ satisfies $\lim_{n\to\infty} μ_n/\log n=1/2$. (2) Let $λ_n$ and $C_n$ denote the infima of the Lipschitz constants of homeomorphisms from $S_{\ell_\infty^n}$ onto $S_{\ell_2^n}$ and from $B_{\ell_\infty^n}$ onto $B_{\ell_2^n}$, respectively. Then $\lim_{n\to\infty} λ_n/\log n=\lim_{n\to\infty} C_n/\log n=1/2$. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.

math.FA↗

arXiv2026

Sphere version of Banach--Kadec--Paley theorem

We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical $L_p$-spaces. Our main result is a sphere version of the classical Banach--Kadec--Paley theorem: for $1\le p,q<\infty$, the unit sphere $S_{L_q}$ admits an almost Lipschitz embedding into $L_p$ if and only if $L_q$ admits a linear embedding into $L_p$. Consequently, the almost Lipschitz geometry of $L_p$-spheres completely determines the underlying linear structure of $L_p$-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton--Randrianarivony's concentration inequalities, Mendel--Naor's metric cotype inequality, and Naor's sharp metric $X_p$ inequality. They also yield several sharp results on the Hölder geometry of $L_p$-spheres.

math.FA↗

arXiv2026

Sharp Metric Cotype Inequalities for $L_1$ via Nonlinear Cut Smoothing

For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metric cotype inequality holds at the sharp scale $m=O(n^{1/q})$. In particular, the quadratic inequality holds with $m=O(\sqrt{n})$, answering the $L_1$ sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.

math.FA↗

arXiv2026

The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature

We determine the optimal order of the scaling parameter in the metric cotype $2$ inequality for complete Alexandrov spaces of nonnegative curvature. It grows linearly with the dimension $n$ of the discrete torus. This answers Question 17 of Eskenazis, Mendel, and Naor. In their sign-vector normalization on $\bigl(\mathbb{Z}/(2m\mathbb{Z})\bigr)^n$, the square of the optimal metric cotype $2$ constant for this class lies between $\max\{1,n/m\}$ and $e\lceil n/m\rceil$ for every dyadic $m\geq 2$. The upper bound follows from a dyadic Bernoulli-thinning argument using only the Lang--Schroeder--Sturm inequality. Snowflake universality of the fixed Wasserstein space $\mathcal{P}_2(\mathbb{R}^3)$ gives the $n/m$ lower bound.

math.FA↗

arXiv2026

Kazhdan's Property $(T)$ for Subspaces and Quotients of $L_p$-Spaces

We prove a uniform spectral-gap estimate at every even exponent for isometric representations on arbitrary closed subspaces of real $L_p$-spaces. As an application, this result, combined with the work of Bader, Furman, Gelander and Monod, gives an affirmative answer to their Question 4.1: Kazhdan's property $(T)$ implies property $(T_X)$ whenever $X$ is a closed subspace or a quotient of a real $L_p$-space, for every $1<p<\infty$.

math.FA↗

arXiv2013

On universal left-stability of $ε$-isometries

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $γ>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leqγ\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that if a dual Banach space $X$ is universally-left-stable, then it is isometric to a complemented $w^*$-closed subspace of $\ell_\infty(Γ)$ for some set $Γ$, hence, an injective space; and that a Banach space is universally-left-stable if and only if it is a cardinality injective space; and universally-left-stability spaces are invariant.

math.FA↗