Faculty Mentor: Chris Connell
For a curve c in Rn, its convex hull is the smallest convex set containing c. There are a number of interesting questions one can ask about the relationship of a curve c to its convex hull. Many of these fall into the mold of an "isoperimetric" type problem where one geometric quantity is being optimized subject to another geometric constraint. A number of authors, including I. Schoenberg and A. Weil, have made important contributions in this area. We will focus on one such question that was related to me by M. Ghomi in 2004. Namely, whether a round planar circle maximizes the surface area of the convex hull among all closed curves c in R3 of a fixed length L. (Here the surface area of the degenerate convex hull of a round circle should be interpreted as twice the area of the disk.) Our first plan will be to attack this problem from an existing approach built upon prior work of N. Krabbenhoft and myself. As is often the case, there are a number of analogues to this question in other settings and in higher dimensions which we can also explore depending on student interest.