Carl W. Lee

My research tends to center around convex polyhedra. These are sets that can be described as intersections of finite numbers of closed halfspaces in Euclidean space. A bounded polyhedron is a convex polytope; equivalently, a convex polytope is the smallest convex set containing a given finite set of points.

My interest in polyhedra is motivated by:

• Their intrinsically interesting properties, such as symmetry (e.g., the Platonic and Archimedean solids, and regular polytopes in higher dimensions), combinatorial properties (e.g., Euler's relation V-E+F=2 for three-dimensional polytopes and extensions and characterizations of face numbers in higher dimensions, paths in the graph determined by a polytope's edges), and metrical properties (e.g., formulas for volume, numbers of interior lattice points, rigidity, triangulations).
• The tools from other areas of mathematics that can be brought to bear on problems on polyhedra, such as commutative algebra, algebraic geometry, enumerative combinatorics, and linear algebra. Some very striking combinatorial results, for example, so far admit no purely combinatorial proof, but require an algebraic perspective.
• The applications of polyhedra to other areas of mathematics, such as linear programming, combinatorial optimization, computational geometry, algebraic geometry, and convexity. For example, in combinatorial optimization one often wishes to optimize a linear functional over a finite (often large) set of points. Frequently one can approach such problems by taking the convex hull of these points, considering an alternate representation of this polytope as an intersection of halfspaces, and exploiting its combinatorial/geometric structure to develop efficient algorithms.

My graduate students are presently working on projects involving stress and toric h-vectors of convex polytopes; triangulations of convex polytopes; and three-dimensional aperiodic tilings.

Some recent publications:

• C.W. Lee, P.L.-spheres, convex polytopes, and stress, Discrete and Computational Geometry 15 (1996) 389-421.
• C.W. Lee, Subdivisions and triangulations of polytopes, Chapter 14 in Handbook of Discrete and Computational Geometry, J.E. Goodman and J. O'Rourke, eds., CRC Press LLC, Boca Raton, 1997, pp. 271-290.