Convex Optimization · Problem 2.21

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · set-of-separating-hyperplanes

Problem

The set of separating hyperplanes. Suppose that

C

and

D

are disjoint subsets of

\mathbf{R}^n

. Consider the set of

(a, b) \in \mathbf{R}^{n+1}

for which

a^T x \leq b

for all

x \in C

, and

a^T x \geq b

for all

x \in D

. Show that this set is a convex cone (which is the singleton

\{0\}

if there is no hyperplane that separates

C

and

D