Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface. It is named after Joseph Liberman.
Formulation If γ {\displaystyle \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 ↦ dist 2 ∘ γ ( t ) − t 2 {\displaystyle 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.