So here is a fact that Vapnik uses to show how the VC dimension of classifiers with a particular margin depends on the margin - he doesn't prove it though, and denotes the proof as using "symmetric considerations", making it seem rather easy. I'm not sure though how to prove it rigorously and elegantly:
Assume that $X$ is an arbitrary set of points $x_1,\dots,x_r$ with diameter $D=\min_{x^\star} \max_{i} \|x_i-x^\star\|_2$ that we can shatter using linear decision rules. Let $T_i(X)$ for $i=1,\dots,2^r$ denote all possible labelings on $X$. Each $T_i(X)$ defines a subdivision of the points into two sets of vectors. We denote the distance between the convex hulls of both sets as $\gamma(T_i(X))$ (can be viewed as the "margin" when thinking about classification problems). Let $X'= \{x'_i\}_{i=1}^r$ be the set of $r$ points located at the vertices of a regular $(r-1)$-simplex inscribed in a sphere of radius $D/2$ that has maximal area among all such simplices. By definition of a regular simplex, the vertices all have the same pairwise distance to one another.
How can I show that $\min_i \gamma(T_i(X)) \leq \min_i \gamma(T_i(X'))$? I'm looking for an elegant/intuitive enough geometric proof rather than writing things down brute-force as an optimization problem etc.
I'm assuming there is some geometric result saying that regular simplex maximizes volume of simplices inscribed in a sphere. Is there perhaps some result relating the volume of a polytope to the minimum distance?