I came up with this question after playing around with some simple geometric arguments:
Let's say that a convex figure is any convex set of points of $\mathbb{R}^2$, with topological boundary any simple closed arc.
Suppose we have two convex figures $A$ and $B$, such that $B\subset A$.
If $L_A, L_B$ are the lengths of their respective boundaries then I have the feeling that $L_A \ge L_B$. Is this true or do I have an mal illusion here? I have tried to prove it by the Green's theorem but convexivity is out of hand here. Thanks for any advise.
EDIT
After this usefull link link of @Eevee Trainer, I realised that nearest-point projection map $p_A$ is the key to prove that $L_A \ge L_B$. But I still have some concerns:
Does any simple closed arc $C$ have length $L_C=\sup \{\text{length of any polygon}\quad X_1 X_2\ldots X_n, \text{ for any division } X_1, X_2,\ldots, X_n \text{ of the arc } C \}$ ?
In order to prove that $p_A$ is onto arc $B$, we need to assume further that the points of $B$ that can have a tangent line is dense on the arc. In that case the rectilinear to the point $y$ meets arc $A$ to a point $x$, and then $p_A(x)=y$. So is it true for a simple closed arc?
Thanks again for any feedback.
