  Let $X$ be a configuration of $n$ points in $\mathbb{R}^d$.
  What is the maximum number of vertices that $conv(T(X))$ can have among all
the possible permissible projective transformations $T$?
  In this paper, we investigate this and connected questions. After presenting
several upper bounds, we study a closely related problem (via Gale transforms)
concerning the number of minimal Radon partitions of a set of points. We then
present some bounds for this number that enable us to partially answer a
question due to Pach and Szegedy. We also discuss another related problem
concerning the size of topes in arrangements of hyperplanes.
