# Documentation

Mathlib.Analysis.Convex.Uniform

# Uniformly convex spaces #

This file defines uniformly convex spaces, which are real normed vector spaces in which for all strictly positive ε, there exists some strictly positive δ such that ε ≤ ‖x - y‖ implies ‖x + y‖ ≤ 2 - δ for all x and y of norm at most than 1. This means that the triangle inequality is strict with a uniform bound, as opposed to strictly convex spaces where the triangle inequality is strict but not necessarily uniformly (‖x + y‖ < ‖x‖ + ‖y‖ for all x and y not in the same ray).

## Main declarations #

UniformConvexSpace E means that E is a uniformly convex space.

## TODO #

• Milman-Pettis
• Hanner's inequalities

## Tags #

convex, uniformly convex

class UniformConvexSpace (E : Type u_1) :

A uniformly convex space is a real normed space where the triangle inequality is strict with a uniform bound. Namely, over the x and y of norm 1, ‖x + y‖ is uniformly bounded above by a constant < 2 when ‖x - y‖ is uniformly bounded below by a positive constant.

Instances
theorem exists_forall_sphere_dist_add_le_two_sub (E : Type u_1) {ε : } (hε : 0 < ε) :
δ, 0 < δ ∀ ⦃x : E⦄, x = 1∀ ⦃y : E⦄, y = 1ε x - yx + y 2 - δ
theorem exists_forall_closed_ball_dist_add_le_two_sub (E : Type u_1) {ε : } [] (hε : 0 < ε) :
δ, 0 < δ ∀ ⦃x : E⦄, x 1∀ ⦃y : E⦄, y 1ε x - yx + y 2 - δ
theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (E : Type u_1) {ε : } [] (hε : 0 < ε) (r : ) :
δ, 0 < δ ∀ ⦃x : E⦄, x r∀ ⦃y : E⦄, y rε x - yx + y 2 * r - δ