---
tag: GDTKW9KPH8
kind: concept
name: Strictly convex space
msc: [46B20]
specializes: [03J5RAKB30]
sources:
  - url: https://en.wikipedia.org/wiki/Strictly_convex_space
same_as:
  wikidata: Q916093
  wikipedia: Strictly convex space
---

## Summary

A normed space is strictly convex when its unit sphere contains no line segment.
Equivalently, the midpoint of two distinct unit vectors has norm less than $1$.
Strict convexity gives uniqueness results, such as uniqueness of the nearest point of a convex set and equality cases in the triangle inequality.
It is weaker than [[uniform convexity|BEAMC5011M]].

## Explanations

### L1 human

Look at the boundary of the unit ball of a normed space.
In a strictly convex space this boundary is curved everywhere and has no flat pieces.
The Euclidean disc is strictly convex.
A square is not, because its sides are flat segments.