Properties of a curve whose convex hull covers a given convex body

Abstract

In this note, we prove the following inequality for the norm of a convex body $K$ in ${\mathbb{R}}^n$, $n\ge 2$: $N(K)\le \frac{{\pi }^{\frac{n-1}{2}}}{2\Gamma \left(\frac{n+1}{2}\right)}\cdot \mathrm{length}(\gamma )+\frac{{\pi }^{\frac{n}{2}-1}}{\Gamma \left(\frac{n}{2}\right)}\cdot \mathrm{diam}(K)$, where $\mathrm{diam}(K)$ is the diameter of $K$, $\gamma$ is any curve in ${\mathbb{R}}^n$ whose convex hull covers $K$, and $\Gamma$ is the gamma function. If in addition $K$ has constant width $\Theta$, then we get the inequality $\mathrm{length}(\gamma )\ge \frac{2(\pi -1)\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\pi }\Gamma \left(\frac{n}{2}\right)}\cdot \Theta \ge 2(\pi -1)\cdot \sqrt{\frac{n-1}{2\pi }}\cdot \Theta$. In addition, we pose several unsolved problems.

arXiv comment7 pages