Theorems · Theorem · general topology

Convexity.dist_convexCombPair_convexCombPair

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

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

Defined in

Mathlib.Analysis.Convex.MetricSpace

Cited by

1 results in Mathlib

Foundations

Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Convexity.ConvexSpaceMetricSpaceConvexity.IsConvexDist