COMPARISON THEOREMS FOR THE VOLUMES OF TUBES ABOUT METRIC BALLS IN CAT(·)-SPACES

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curva-ture. First we compare the Hausdor？？ measure of tube about a metric ball contained in an (n ¡ 1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically com-plete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space Rn, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with additional condition. Also, we estimate the Hausdor？？ measure of the tube about a shortest curve in a metric space of curvature bounded above and below.