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

In this note, we prove the following inequality for the norm of a convex body $K$ in ${\mathbb{R}}^n$, $n\ge 2$: $N(K) \leq \frac{\pi^{\frac{n-1}{2}}}{2 \Gamma \left(\frac{n+1}{2}\right)}\cdot \operatorname{length} (\gamma) + \frac{\pi^{\frac{n}{2}-1}}{\Gamma \left(\frac{n}{2}\right)} \cdot \operatorname{diam}(K)$,where$\operatorname{diam}(K)$isthediameterof$K$,$\gamma$isanycurvein$$\mathbb{R}$^n$whoseconvexhullcovers$K$,and$\Gamma$isthegammafunction\mathrm{.}Ifinaddition$K$hasconstantwidth$\Theta$, 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.