  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.
