Theorems · Theorem · functional analysis

StrictConvexSpace.of_norm_combo_lt_one

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E],
  (∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1) → StrictConvexSpace ℝ E

Strict convexity is equivalent to ‖a • x + b • y‖ < 1 for all x and y of norm at most 1 and all strictly positive a and b such that a + b = 1. This lemma shows that it suffices to check this for points of norm one and some a, b such that a + b = 1.

Defined in

Mathlib.Analysis.Convex.StrictConvexSpace

Cited by

0 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace