Short map - Definition and Overview

In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any <math>x,y\in X<math> we have

<math>d_Y(f(x),f(y)) \le d_X(x,y)<math>.

Here <math>d_X<math> and <math>d_Y<math> denote metrics on <math>X<math> and <math>Y<math>, respectively. In other words, f is short iff it is 1-Lipschitz.

One can say that f is strictly short if the inequality is always strict. Then a contraction mapping is strictly short, but not conversely (even with X = Y).