Total curvature of graphs in Euclidean spaces (Q1127858)

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.