Some Extremal Questions for Simplicial Complexes. I: Polyhedral Geodesic Strips
Abstract
In this first note, we are concerned with an inequality between the area of a strip and the product of two lengths. A corresponding local inequality occurs in unpublished work of Aronszajn and Choquet, which is now quite old, and fairly well-known in its main features. We establish a polyhedral form of our inequality in the large, and we note that it necessitates modifying the local inequality by a constant factor, but we do not obtain the sharpest value for this factor. Our result extends without serious difficulty to relatively elementary curved surfaces,such as curved polyhedra and differentiable Z-manifolds with boundary, to which our methods apply virtually unaltered. For a more general extension, the apparatus would need modifying in a manner studied by K. H. Carlson and by R. W. Rishel in their theses [1,2]. The variational purpose of our inequality originates with [3]. Some connected questions are listed at the end of this note and in note II.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1962
- Accession Number
- AD0298806
Entities
People
- L. C. Young
Organizations
- University of Wisconsin–Madison