Polytopal balls arising in optimization

Umbrellas and polytopal approximation of the euclidean ball

There are two positive, absolute constants $c_{1}$ and $c_{2}$ so that the volume of the difference set of the $d$-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than $$ c_{2}\ d\ {n}^{-\frac{2}{d-1}}vol_d(B^d_2) $$ for $n \geq (c_{1}\ d)^{\frac{d-1}{2}}$.

Funk几何的体积增长与多胞形的旗

Volume growth of Funk geometry and the flags of polytopes

本文研究Funk几何中球的体积增长，提出并部分验证了与Mahler和旗猜想相关的极小化猜想，并计算了多胞形情形的高阶渐近项，推广了Santaló点概念。

凸体中的随机多面体：弥合极值容器之间的差距

Random polytopes in convex bodies: Bridging the gap between extremal containers

本研究探讨块β分布族中随机点凸包构成的随机多面体的渐近性质，建立面期望数量的显式增长率，分析均匀分布下任意维度面的期望数量及体积差，揭示其插值性质以弥合两类凸容器中随机多面体行为的差距。

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \rightarrow V$ is a nonexpansive map on a finite dimensional normed space $V$, then the fixed point set of $f$ is nonempty and bounded if and only if there exist $w_1, \ldots , w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss application…