Theorems · Theorem · general topology

Convexity.dist_convexCombPair_convexCombPair_le

∀ {X : Type u_2} [inst : Convexity.ConvexSpace ℝ X] [inst_1 : MetricSpace X] [Convexity.IsConvexDist X] {s t : ℝ}
  (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y x' y' : X),
  dist (Convexity.convexCombPair s t hs ht h x y) (Convexity.convexCombPair s t hs ht h x' y') ≤
    s * dist x x' + t * dist y y'

dist(sx + (1-s)y, sx' + (1-s)y') ≤ s dist(x, x') + (1-s) dist(y, y'). See dist_convexCombPair_convexCombPair for the version where the points are fixed and the weights change.

Defined in

Mathlib.Analysis.Convex.MetricSpace

Cited by

2 results in Mathlib

Foundations

Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Convexity.ConvexSpaceMetricSpaceConvexity.IsConvexDist