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?



Inequality $\frac{C}{d_{max}} \le \pi $ relating perimeter and diameter of planar convex body
Bonnesen and Fenchel’s *Theorie der konvexen Körper* ([1934](//zbmath.org/?q=an:60.0673.01), [(6) p. 77](//books.google.com/books?id=cvGoBgAAQBAJ&pg=PA77)) ([translation](//zbmath.org/?q=an:0628.52001)) 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](//zbmath.org/?q=an:51.0593.08)), with the planar case $n=2$ already in Blaschke ([1915](https://zbmath.org/?q=an:45.0731.03)) and Rosenthal-Szász ([1916](//zbmath.org/?q=an:46.1119.03)).


