Tight Bounds for Approximate Carathéodory and Beyond

Subgradients. A function f : Q ✓ R ! R defined on a convex domain Q is said to be convex if every point x 2 Q has a non-empty subgradient @f(x) = {g 2 R; f(y) f(x) + g>(y x), 8y 2 Q}. Geometrically, this means that a function is convex iff it is the maximum of all its supporting hyperplanes, i.e. f(x) = max x0,g2@f(x0) f(x0) + g > (x x0). When there is a unique element in @f(x) we call it the gradient and denote it by rf(x). We will sometimes abuse notation and refer to rf(x) as an arbitrary element of @f(x) even when it is not unique. Strong convexity and smoothness. We say that a function f : Q ✓ R ! R is μ-strongly convex with respect to norm k·k if for all x, y 2 Q and all subgradients g 2 @f(x): f(y) f(x) g>(y x) 1