UC Davis Mathematics

(George) Thomas Sallee

Professor Emeritus
Convex geometry
Ph.D., 1966, University of Washington
Refereed publications: Via Math Reviews

Email: sallee@math.ucdavis.edu
Office: MSB 2236
Phone: 752-2212

Professor G. Thomas Sallee studies convex geometry. He has solved a number of problems with elementary statements involving common geometric objects and constructions such as Reuleaux triangles, Euler characteristic, and convex bodies.

A Reuleaux triangle is the simplest example of a convex body with constant width, a property that implies that the body could arise as the shape of a ball bearing or a rolling pin supporting a flat surface in a mechanism. (For other reasons, it also appears in rotary engines.) In a series of papers, Professor Sallee investigated various properties of convex bodies of constant width, pairs of constant relative width, and Reuleaux polytopes [2] [4]. In particular, he showed that any convex body of constant width in any dimension is a limit of Reuleaux polytopes. This is an example of a finite approximation theorem, analogous to the foundational results that every smooth function can be approximated by polynomials (the Weierstrass approximation theorem) and that every convex body can be approximated by ordinary polytopes (which is important in linear programming).

In another investigation, Professor Sallee demonstrated that a valuation on convex polytopes is the sum of a function satisfying a positive Euler relation and a negative Euler relation [1]. Each of these concepts provides an axiomatic framework for various statistics of convex polytopes, including the total number of faces and the Euler characteristic. This paper motivated the problem of characterizing all functions on polytopes that satisfy an Euler relation, which led to an extended research topic for several authors and culminated after 20 years in a complete solution by McMullen.

Professor Sallee has also devoted careful attention to math education. For example, Kasimatis and Sallee designed an influential curriculum to prepare junior high school and high school students for college mathematics [5].

Selected publications

[1] Polytopes, valuations and the Euler relation, Canad. J. Math. 20 (1968), 1412-1424.

[2] Reuleaux polytopes, Mathematika 17 (1970), 315-323.

[3] Tiling convex sets by translates, Israel J. Math. 14 (1976), 368-376.

[4] Pairs of sets of constant width, J. Geom. 29 (1987), 1-11.

[5] Designing a College Preparatory Mathematics Course: An Overview (with Elaine Kasimatis), CBMS Issues in Mathematics Education, American Math. Society 3 (1993), 81-102.

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