TITLE: A convex curve inside the unit circle
QUESTION [6 upvotes]: Has this theorem a specific name; and I need some references for general form. 
Suppose ${\gamma}$ is a convex polygon contained in the interior of the unit circle. then the length of $\gamma$ is not more than the length of the unit circle.

REPLY [3 votes]: In general (say in $\mathbb R^n$) the perimeter of a set $E$ containing a convex set $C$ is not smaller than the perimeter of $C$.
This is a consequence of the following facts about the projection $\pi$ on the convex set $C$:

$\pi(\partial E) \supset \partial C$
$\pi$ is a non-expanding map (i.e. $1$-lipschitz)
the surface area $\mathcal H^{n-1}$ of a non-expanding image of a set is not larger than the surface area of the set.