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.