Mathematics — Graph Theory · piecewise-linear-paths-star-shaped
Problem
A subspace X⊂Rn is said to be star-shaped if there is a point x0∈X such that, for each x∈X, the line segment from x0 to x lies in X. Show that if a subspace X⊂Rn is locally star-shaped, in the sense that every point of X has a star-shaped neighborhood in X, then every path in X is homotopic in X to a piecewise linear path, that is, a path consisting of a finite number of straight line segments traversed at constant speed. Show this applies in particular when X is open or when X is a union of finitely many closed convex sets.