Abstract
Convexity plays an essential role in many areas of mathematics, from geometry, analysis and linear algebra to numerous applications in other areas of mathematics such as optimization. It unifies many apparently diverse mathematical phenomena, and is relevant to engineering and the sciences. In practice, however, convexity does not always hold, which raises the need for suitable generalizations of convexity. In this thesis, I will study generalizations of convexity and use metric entropy to give a numerical quantification of the collections of sets and function classes which satisfy these generalized convexity conditions. In particular, I will obtain the metric entropy estimate of the collection of bounded sets in $\R^d$ with positive reach, the metric entropy of an $l_q$-hull ($0<q\le 1$) in an important case, as well as the upper bound for the metric entropy of separately convex functions in $\R^d$.