Title: Symmetry Functions and Symmetry Points of a Convex Set: Properties, Duality, and Computational Complexity Given a closed convex set C and a point x \in C, let sym(x,C) measure how symmetric C is about the point x, and define sym(C) to be the largest value of sym(x,C) over all x \in C. We call x^s a symmetry point of C if x^s achieves the above supremum. These symmetry measures are all invariant under linear transformation and/or change in norm, and so are of interest in the study of the geometry of convex sets. Furthermore, these measures arise naturally in the complexity theory of interior-point methods. We show various properties of sym(x,C) under duality correspondences between C and C^*, and study the problem of finding a projective transformation T( .) of C that maximizes sym(T(C)). When C is a polytope given by the intersection of m halfspaces, sym(C) can be computed by solving a certain linear program in m^2 variables, whose computational complexity depends only on sym(C), m, and the desired relative accuracy of solution.