Convex Optimization · Problem 2.1

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · definition-of-convexity

Problem

Let

C \subseteq \mathbf{R}^n

be a convex set, with

x_1, \ldots, x_k \in C

, and let

\theta_1, \ldots, \theta_k \in \mathbf{R}

satisfy

\theta_i \geq 0

\theta_1 + \cdots + \theta_k = 1

. Show that

\theta_1 x_1 + \cdots + \theta_k x_k \in C

. (The definition of convexity is that this holds for

k = 2

; you must show it for arbitrary

k

.) Hint. Use induction on

k