## Document Type

Dissertation

## Date of Award

1975

## Keywords

Simplexes (Mathematics)

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematical Sciences

## First Advisor

Prabir Roy

## Second Advisor

Louis F. McAuley

## Third Advisor

James Geer

## Abstract

If V is a finite set of points {V_{1}, V_{2}, …, V_{r}} in some Euclidean space, E^{n}, then the convex closure, C, of these points is a convex linear cell in E^{n} whose vertex set is some subset of V. Given a (finite) collection of convex linear cells, each specified by its vertex set, then the intersection of these cells is again a cell with a unique set of vertices. However, to efficiently determine which of {V_{1}, …, V_{r}} are vertices (or non-vertices) of C, or to determine the vertex set of the intersection from the vertex sets of the intersecting cells is not, in general, a trivial matter. Figure 1 shows, for example, that the effect of “small movements” of individual cells, upon the dimension of the cell of intersection or upon the number of vertices of this cell of intersection depends on how the movement is applied to the vertices of the original cells.

A partial solution to the problem of determining the vertices of the intersection from the vertices of the intersecting cells was obtained by the author as a Master's thesis problem. It was shown that if σ_{1}, σ_{2}, ...,σ_{k} are simplices in E^{n}, then the vertex set of ^{k}∩_{1} σ_{i} may be obtained by examining all maximal nonsingular submatrices of a matrix obtained from the coordinates of the vertices of each of σ_{1}, σ_{2}, …, σ_{k}. It is hoped that insight into the general problem of how vertices of individual cells determine the vertices of the intersection and how this intersection is affected by small movements of vertices may be obtained via the framework developed herein.

## Recommended Citation

Thomas, David Robert, "Symmetric products of cubes" (1975). *Graduate Dissertations and Theses*. 295.

https://orb.binghamton.edu/dissertation_and_theses/295