Thesis (Ph. D.)--University of Michigan. / Lithoprinted. Thesis note mounted on t.p.

Thesis (Ph. D.)--University of Michigan. / Lithoprinted. Thesis note mounted on t.p.

Thesis (M. S.)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Evans Harrell; Committee Member: Guillermo Goldsztein; Committee Member: Mohammad Ghomi

This thesis examines the elegant theory of polar and Legendre duality, and its potential use in convex geometry and geometric analysis. It derives a theorem of polar - Legendre duality for all convex bodies, which is captured in a commutative diagram. A geometric flow on a convex body induces a distortion on its polar dual. In general these distortions are not flows defined by local curvature, but in two dimensions they do have similarities to the inverse flows on the original convex bodies. These ideas can be extended to higher dimensions. Polar - Legendre duality can also be used to examine Mahler's Conjecture in convex geometry. The theory presents new insight on the resolved two-dimensional problem, and presents some ideas on new approaches to the still open three dimensional problem.