Let $C$ is a perimeter of a convex hull (plane geometry) and $d_{max}$ is the largest distance of two arbitrary points in the convex hull. I am looking for a proof that:
$$\frac{C}{d_{max}} \le \pi $$
What is a generalization of the inequality for higher dimension?
Bonnesen and Fenchel’s Theorie der konvexen Körper (1934, (6) p. 77) (translation) gives the generalization to a convex body $K\subset\mathbf R^n$ as $$ \mathrm{vol}(\partial K)\leqslant \omega_n\left(\dfrac{\mathrm{diameter}(K)}2\right)^{n-1} $$ where $\omega_n=\mathrm{vol}(S^{n-1})=2\pi^{n/2}\,/\,\Gamma(\frac n2)$. They seem to attribute it (p. 107) to Kubota (1925), with the planar case $n=2$ already in Blaschke (1915) and Rosenthal-Szász (1916).
