Most Finite Point Sets in the Plane have Dilation $$>1$$ > 1

Abstract

We prove the following fact for arbitrary finite point sets $$S$$S in the plane. Either, S is a subset of one of the well-known sets of points whose triangulation is unique and has dilation 1. Or there exists a number $$\Delta (S) > 1$$Δ(S)>1 such that each finite plane graph containing $$S$$S among its vertices has dilation $$\ge \Delta (S)$$?Δ(S).