Uniformly convex spaces #

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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 #

uniform_convex_space E means that E is a uniformly convex space.

TODO #

Milman-Pettis
Hanner's inequalities

Tags #

convex, uniformly convex

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@[class]

structure uniform_convex_space (E : Type u_1) [seminormed_add_comm_group E] :

Prop

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)

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.

See also uniform_convex_space.of_uniform_convex_closed_unit_ball.

Instances of this typeclass

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theorem exists_forall_sphere_dist_add_le_two_sub (E : Type u_1) [seminormed_add_comm_group E] [uniform_convex_space E] {ε : ℝ} (hε : 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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theorem exists_forall_closed_ball_dist_add_le_two_sub (E : Type u_1) [seminormed_add_comm_group E] [uniform_convex_space E] {ε : ℝ} [normed_space ℝ E] (hε : 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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theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (E : Type u_1) [seminormed_add_comm_group E] [uniform_convex_space E] {ε : ℝ} [normed_space ℝ E] (hε : 0 < ε) (r : ℝ) :

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

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@[protected, instance]

def uniform_convex_space.to_strict_convex_space {E : Type u_1} [normed_add_comm_group E] [normed_space ℝ E] [uniform_convex_space E] :

strict_convex_space ℝ E