 
Summary: EXTRINSIC CURVATURE OF SEMICONVEX SUBSPACES IN
ALEXANDROV GEOMETRY
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. In Alexandrov spaces of curvature bounded either above (CBA)
or below (CBB), we obtain extrinsic curvature bounds on subspaces associ
ated with semiconcave functions. For CBA spaces, we obtain new intrinsic
curvature bounds on subspaces. For CBB spaces whose boundary is extrin
sically curved, we strengthen Perelman's concavity theorem for distance from
the boundary, deriving corollaries on sharp diameter bounds, contractibility,
and rigidity.
Contents
1. Introduction 1
2. Almost concave functions 6
3. Very concave functions 12
4. Curved boundaries 15
5. Baseangle/chord and arc/chord curvature bounds 20
Acknowledgments 22
References 22
1. Introduction
Semiconcave functions arise naturally in Alexandrov geometry, and are power
