Total curvature of graphs in Euclidean spaces

Abstract

In this paper we define the total curvature of a polygonal map from a finite graph G to a Euclidean space E n . We characterize for certain G the polygonal maps with minimal total curvature. When G is homeomorphic to a circle the result is the piecewise linear version of the generalized Fenchel theorem on the total curvature of a smooth closed curve in a Euclidean space.