Liberman's lemma

Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface. It is named after Joseph Liberman.

Formulation

If $\gamma$ is a unit-speed minimizing geodesic on the surface of a convex body K in Euclidean space then for any point p ∈ K, the function

$t\mapsto\operatorname{dist}^2\circ\gamma(t)-t^2 \,$

is concave.

References

• Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.