arXiv subjects

Chan He

Publications and source records attributed to Chan He.

3 recordsLinked to original sources

arXiv2026

Dimension-free estimates for covering functionals of simplices and $\ell_p$ balls

We study $\Gamma_{2^n}(K)$, the least positive number $\gamma>0$ such that an $n$-dimensional convex body $K$ can be covered by $2^n$ translates of $\gamma K$. For $n$-simplices $\Delta_n$, we prove that $\Gamma_{2^n}(\Delta_n)$, as a sequence in $n$, tends to $1/2$. For the cross-polytope $B_1^n$, we show that $\Gamma_{2^n}(B_1^n)\leq5/6$ holds for all $n\geq2$, and that $\limsup_{n\to\infty}\Gamma_{2^n}(B_1^n)\leq0.641\cdots$. Finally, we prove the existence of a constant $\kappa_*<1$ such that $\Gamma_{2^n}(B_p^n)\leq\kappa_*$ for all $n\geq2$ and all $p\in[1,\infty]$.

math.MG↗

arXiv2021

Homothetic covering of convex hulls of compact convex sets

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $\lambda\in[0,1]$ such that $K$ can be covered by $m$ translates of $\lambda K$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.

math.MG↗

arXiv2018

Uniqueness of completions and related topics

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set $K$ having the same diameter as $K$ is called a completion of $K$. In general, a bounded set may have different completions. We study normed linear spaces having the property that there exists a nontrivial segment with a unique completion. It turns out that this property is strictly weaker than the property that each complete set is a ball, and it is strictly stronger than the property that each set of constant width is a ball. Extensions of this property are also discussed.

math.FA↗