---
tag: BEAMC5011M
kind: concept
name: Uniformly convex space
msc: [46B20]
specializes: [GDTKW9KPH8]
sources:
  - url: https://en.wikipedia.org/wiki/Uniformly_convex_space
same_as:
  wikidata: Q847535
  wikipedia: Uniformly convex space
---

## Summary

A normed space is uniformly convex when midpoints of two unit vectors that are far apart lie uniformly deep inside the unit ball.
It is a quantitative form of [[strict convexity|GDTKW9KPH8]] of the unit ball.
Hilbert spaces and $L^p$ spaces for $1 < p < \infty$ are uniformly convex, and the [[Milman–Pettis theorem|AJ5RG9XQYZ]] says that every uniformly convex Banach space is reflexive.

## Explanations

### L1 human

Picture the unit ball as a round ball.
If two points on or in the ball are a fixed distance apart, the midpoint of the segment between them is noticeably inside the ball, not near its edge.
Uniform convexity asks that this happen with a margin that depends only on the distance between the points.
A square ball in the plane fails this: the midpoint of two corners on the same side lies on the boundary.