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

TL;DR

This paper establishes inequalities relating the norm of a convex body to the length of curves covering it, with special cases for bodies of constant width, and discusses open problems in the geometry of convex bodies.

Contribution

It introduces new inequalities connecting convex body properties with covering curves and explores the case of bodies with constant width, proposing several open problems.

Findings

Derived an inequality for the norm of convex bodies involving covering curve length and diameter.

Established a lower bound on the length of covering curves for bodies with constant width.

Proposed several open problems related to convex body coverings and properties.

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.