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 $\mathbb{R}^n$, $n\geq 2$: $N(K) \leq \frac{π^{\frac{n-1}{2}}}{2 Γ\left(\frac{n+1}{2}\right)}\cdot \operatorname{length} (γ) + \frac{π^{\frac{n}{2}-1}}{Γ\left(\frac{n}{2}\right)} \cdot \operatorname{diam}(K)$, where $\operatorname{diam}(K)$ is the diameter of $K$, $γ$ is any curve in $\mathbb{R}^n$ whose convex hull covers $K$, and $Γ$ is the gamma function. If in addition $K$ has constant width $Θ$, then we get the inequality $\operatorname{length} (γ) \geq \frac{2(π-1)Γ\left(\frac{n+1}{2}\right)}{\sqrtπ\,Γ\left(\frac{n}{2}\right)}\cdot Θ\geq 2(π-1) \cdot \sqrt{\frac{n-1}{2π}}\cdot Θ$. In addition, we pose several unsolved problems.