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

Abstract

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n − 1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete 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 an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

This article was migrated from the previous system via automation. The abstract may not be written correctly. Please view the PDF file.