度量测度空间中球的渐近等周性

延拓、分离与同构反向等周

Extension, separation and isomorphic reverse isoperimetry

本文研究了度量空间的Lipschitz延拓模与分离模，通过有限维赋范空间的随机聚类几何结构获得了两者的上下界，并首次精确评估了特定赋范空间Lipschitz延拓模的阶。

Schatten球体积的下一阶渐进行为

Next-order asymptotics for the volume of Schatten balls

研究Schatten球体积的渐进行为，通过配分函数渐进行为证明，为一般p>1的情况提供下一阶渐近展开。

Exact asymptotic volume and volume ratio of Schatten unit balls

The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond [Studia Math.\ 80, 63--75, 1984] showed that the limit $$ \lim_{n\to\infty} n^{1/2 + 1/p} \big(\text{Vol}\, B_p^n(\mathbb{R})\big)^{1/n^2} $$ exists in $(0,\infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields.…

The maximum entropy principle and volumetric properties of Orlicz balls

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs prin…

Volume of small balls and sub-Riemannian curvature in 3D contact manifolds

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

关于$\rb^{n}$和Marcinkiewicz空间的等球的重填充

On densest packings of equal balls of $\rb^{n}$ and Marcinkiewicz spaces

我们调查，“一拉Marcinkiewicz”技术应用于（渐近）密度函数，如何稠密系统的相等领域的$\rb^{n}, n \geq 1,$可以在无穷远处分割，以允许计算其密度作为一个真正的限制，而不是limsup。等球堆积的密度是Marcinkiewicz意义下球系的特征函数的范数1。给出了密度球填充和最大密度完全饱和球填充的存在性定理的新的直接证明。

Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodies $L$. We prove the following sharp lower bound $$c \sqrt{n} \leq \mbox{lvr}(K),$$ for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{\frac{n}{\log \log(n)}}$. We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that $\mbox{lvr}(K)$ behaves as the square…

Asymptotic volume in Hilbert Geometries

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric

We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = ρ\cdot (n\!-\!1)$, $ρ\in [0,1]$, i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most $0.029$ bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.