Notes on Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

This is a set of notes that is basically and expanded version of the paper Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls. The differences are a few extra pictures, Section 2.7 which is an exposition of results of Ng and Nikodem [5] about measurable approximately convex functions, and an alternate proof of Theorem 2.27 is included. Contents 1. Introduction 2 2. Approximately Convex Functions 4 2.1. Bounds on approximately convex functions 5 2.2. Lower semi-continuity and mean value properties of extremal approximately convex functions 7 2.3. The extremal approximately sub-affine function H(x) 12 2.4. The extremal approximately convex function E(x) on a simplex 19 2.5. Extremal approximately convex functions on convex polytopes 28 2.6. A stability theorem of Hyers-Ulam type 29 2.7. Measurable approximately convex functions 30 2.8.