Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above

In this paper, we show a local energy convexity of W-1,W-2 maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.