MSc thesis defence in Mathematics

Speaker: Mr. Shardul Vikram

Abstract: The L_p-Minkowski problem, a generalization of the classical Minkowski problem, was defined by Lutwak in the '90s. For a fixed real number p, it asks what are the necessary and sufficient conditions on a finite Borel measure on 𝕊^n-1 so that it is the L_p surface area measure of a convex body in ℝⁿ. For p=1, one has the classical Minkowski problem in which the L_p surface area is the usual surface area of a compact set embedded in ℝⁿ.

Under certain technical assumptions, the planar L_p-Minkowski problem reduces to the study of positive, π-periodic solutions, h: [0, 2π] → (0,¥) to the non-linear equation h^1-p (h˝ + h) = y  for a given smooth function y: [0, 2 π] → (0,¥).

In this thesis, we give a new proof of the existence of solutions of the planar L_p-Minkowski problem for 0 < p < 1. To do so, we consider a parabolic anisotropic curvature flow on the space of strictly convex bodies 𝙺 Î ℝ², which are symmetric with respect to the origin.

The connection between solutions to a parabolic equation, the flow, and a corresponding elliptic equation, the L_p-Minkowski problem, has been long conjectured by the specialists and this is yet another instance where it has been used.