Convex functions: Convex Functions

affine function Banach space Beckenbach boundary point called compact concave constraints converges convex combination convex set differentiable at x0 dual problem equivalent example exists extreme points f is bounded f is continuous f is convex feasible set feasible solution finite finite-dimensional follows Fréchet Fréchet derivative function f functions convex Gateaux differentiable hyperplane implies inner product space interval Jensen's inequality Let f linear functional linear programming linear programming problem linear transformation Lipschitz Math mathematics maximal maximum midconvex function minimize mixed strategies nonempty nonnegative definite normed linear space open set optimal solution permutation matrix Proof properties proved quasiconvex real numbers Rockafellar s f(x saddle point satisfies Section sequence set U C L strictly convex subset subspace Suppose f symmetric tableau Theorem topologically unique unit sphere vector