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

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class UniformConvexSpace (E : Type u_1) [SeminormedAddCommGroup E] :

Prop

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.

uniform_convex : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ

Instances

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theorem UniformConvexSpace.uniform_convex {E : Type u_1} :

∀ {inst : SeminormedAddCommGroup E} [self : UniformConvexSpace E] ⦃ε : ℝ⦄, 0 < ε → ∃ (δ : ℝ), 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ

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