Convex Optimization · Problem 2.26

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · support-function

Problem

Support function. The support function of a set

C \subseteq \mathbf{R}^n

is defined as

S_C(y) = \sup\{y^T x \mid x \in C\}.

(We allow

S_C(y)

to take on the value

+\infty

.) Suppose that

C

and

D

are closed convex sets in

\mathbf{R}^n

. Show that

C = D

if and only if their support functions are equal.