Is the boundary measure ("surface area") of a convex set in $\mathbb{R}^n$ less than the boundary measure of it's enclosing hypersphere (smallest hypersphere that contains the set)?
In 2D I've found a paper that states it's true, and intuitively I think it should be true for n-D, but I'm having trouble proving it.
Edit Possible solution: approximate the boundary of the inner set with panels - line segments/triangles/tetrahedra/ect depending on the dimension. Then orthogonally project those panels onto the hypersphere. Since the set is convex, the projections of the panels don't overlap, and since the projections are orthogonal, their projections onto the sphere are larger than the original panels.
Edit 2 More general conjecture: If $X$ and $Y$ are convex sets with $X \subset Y \subset \mathbb{R}^n$, then $|\partial X| \le |\partial Y|$