The relations between volume ratios and new concepts of gl constants

Volume ratios and a reverse isoperimetric inequality

It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no larger than the corresponding expression for a regular $n$-dimensional ``tetrahedron''. It is also shown that among $n$-dimensional subspaces of $L_p$ (for each $p\in [1,\infty]), \ell^n_p$ has maximal volume ratio.\vskip3in

Centroids and comparison of volumes

For $-1<p<1$ we introduce the concept of a polar $p$-centroid body ${Γ^*_p K}$ of a star body $K$. We consider the question of whether ${Γ^*_p K}\subset {Γ^*_p L}$ implies $\mathrm{vol}(L)\le \mathrm{vol}(K).$ Our results extend the studies by Lutwak in the case $p=1$ and Grinberg, Zhang in the case $p> 1$.

关于度量测度空间中分数阶Sobolev半范数的渐近行为：渐近体积比、体积熵与刚性

On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity

本文研究一般度量测度空间中分数阶Sobolev半范数的渐近行为，给出Maz'ya-Shaposhnikova定理和Ludwig结果的新简短证明，建立渐近公式与渐近体积比之间的联系，并证明两个新的刚性结果。

关于投影外体积比的一个注记

A remark on projection outer volume ratio

本文研究投影外体积比，在一般情形和欧几里得球情形下给出尖锐上界，并证明所得界在相差对数因子下最优。

On the volume ratio of projections of convex bodies

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies $K$ and $L$ is large. Namely, we prove that for every $1\leq k\leq n$ and for each convex body $K\subset \mathbb{R}^n$ there is a centrally symmetric body $L \subset \mathbb{R}^n$ such that for any two projections $P, Q: \mathbb{R}^n \to \mathbb{R}^n$ of rank $k$ one has $$ \mbox{vr}(PK, QL) \geq c \, \min\left\{\frac{ k}{ \sqrt{n}} \, \sqrt{\frac{1}{\log \log…

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.…