Intuition for why a convex set with empty interior lies in an affine set

Question

In Section 2.5.2 of Boyd & Vandenberghe's Convex Optimization, the authors claim that

a convex set in $\mathbb{R}^n$ with empty interior must lie in an affine set of dimension less than $n$.

Can someone provide some intuitive explanations of what this means?

Added to The List. – Rodrigo de Azevedo Apr 15 '21 at 11:16 — Rodrigo de Azevedo, Apr 15 '21 at 11:16

score 1 · Accepted Answer · answered Apr 13 '21 at 05:06

1

A convex set contains the line segment between any two points therein. If it is not contained in an affine subspace, it contains an $n$-simplex.

answered Apr 13 '21 at 05:06

Igor Rivin

1 Answers1