Curvature properties of typical convex surfaces

Tudor Zamfirescu

Vol. 131 (1988), No. 1, 191–207

DOI: 10.2140/pjm.1988.131.191

Abstract

Here we shall see that on typical convex surfaces the set of points with an infinite sectional curvature in some direction and that of points in which the lower sectional curvature in some direction equals the upper sectional curvature in the opposite direction are dense. Also we shall see that, in a certain sense, most convex surfaces are a.e. “very close” to their tangent hyperplane, closer than vanishing curvature already indicates.

Mathematical Subject Classification 2000

Primary: 53C45

Secondary: 52A20, 53A07

Milestones

Received: 3 July 1986

Revised: 3 November 1986

Published: 1 January 1988