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

In this note, we prove the following inequality for norm N(K) of a convex body K in $$\mathbb {R}^n$$ , $$n\ge 2$$ : $$\begin{aligned} \le \frac{\pi ^{\frac{n-1}{2}}}{2 \Gamma \left( \frac{n+1}{2}\right) }\cdot {{\,\mathrm{length}\,}}(\gamma ) + ^{\frac{n}{2}-1}}{\Gamma \frac{n}{2}\right) } \cdot {{\,\mathrm{diam}\,}}(K), \end{aligned}$$ where $${{\,\mathrm{diam}\,}}(K)$$ is diameter K, $$\gamma $$ any curve whose hull covers and $$\Gamma gamma function. If addition has constant width $$\Theta then get \ge \frac{2(\pi -1)\Gamma }{\sqrt{\pi }\,\Gamma \Theta 2(\pi -1) \sqrt{\frac{n-1}{2\pi }}\cdot . addition, pose several unsolved problems.