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 $\mathbb{R}^n$, $n\geq 2$: $N(K) \leq \frac{\pi^{\frac{n-1}{2}}}{2 \Gamma \left(\frac{n+1}{2}\right)}\cdot \operatorname{length} (\gamma) + \frac{\pi^{\frac{n}{2}-1}}{\Gamma \left(\frac{n}{2}\right)} \cdot \operatorname{diam}(K)$, where $\operatorname{diam}(K)$ 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 $\operatorname{length} (\gamma) \geq \frac{2(\pi-1)\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\pi}\,\Gamma \left(\frac{n}{2}\right)}\cdot \Theta \geq 2(\pi-1) \cdot \sqrt{\frac{n-1}{2\pi}}\cdot \Theta$. In addition, we pose several unsolved problems.