---
tag: 62FGRXS27K
kind: concept
name: Strictly convex set
msc: [52A07, 52A20]
specializes: [02AS45T3H9]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_set
---

## Summary

A strictly convex set is a [[convex set|02AS45T3H9]] in which the open segment between any two distinct points lies in the [[interior|0RZH6W98HK]] of the set.
Its boundary contains no flat piece: no nontrivial segment lies on the boundary.
A normed space is a [[strictly convex space|GDTKW9KPH8]] exactly when its closed unit ball is a strictly convex set.
Strict convexity gives uniqueness, for example of the point of a set where a nonzero linear function is largest.

## Explanations

### L1 human

A disc is strictly convex: join two of its points and every point strictly between them is well inside the disc.
A square is convex but not strictly convex.
Two points on the same side give a segment that runs along the edge and never enters the inside.
So strict convexity means that the boundary bulges outward everywhere and has no straight parts.