Liberman's lemma is a theorem used in studying intrinsic geometry of

convex surface In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its ''epigraph ...

. It is named after

Joseph Liberman Joseph Liberman (1917 in Henichesk – August 1941 in ) was a Soviet mathematician, a student of Aleksandrov, best known for Liberman's lemma. Biography In 1936 he entered Leningrad State University as one of the winners of the first city Mat ...

Formulation

\gamma

is a unit-speed minimizing

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

on the surface of a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it ...

''K'' in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

then for any point ''p'' ∈ ''K'', the function :

t\mapsto\operatorname^2\circ\gamma(t)-t^2

is concave.

References

*Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313. Differential geometry of surfaces Lemmas {{differential-geometry-stub