The mixed volume optimization problem

Abstract

The mixed volume optimization problem is to determine the point of duality Q for a given convex set K that minimizes the “mixed volume” of the associated polar set (K ∗;Q) . In the plane, the mixed volumes translate as the area and length; in space, the mixed volumes include the volume, surface area, and mean width. In this paper, the geometric optimization problems associated with minimizing mixed volumes are examined from two perspectives: enumerative search and symbolic computation. The problem of minimizing the polar area through an enumerative search is first considered. The dual polygon (P m ∗;Q) is constructed for an arbitrary point of duality Q∈ P m ° by using an algebraic correspondence between the edges of P m and the vertices of (P m ∗;Q) , and the area of (P m ∗;Q) , A(P ∗ m;Q) , is calculated and minimized using naive search techniques. A result due to Santaló is applied to verify the minimizing solution, and computational tests are described for various classes of randomly generated polygons. Statistical evidence indicates that a “good” approximation to the minimum area polar polygon occurs when the duality point is located at the center-of-gravity of P m . The polar area problem is then investigated using symbolic procedures. Explicit symbolic expressions for the polar area and length functionals are computed and solved directly using the differential optimality conditions and Newton's iterative method of solution. The mixed volume and surface area functionals are formulated and solved using numerical products, and the mean width functional is described. Examples are used throughout to illustrate the methodology.