Abstract

Among the various important aspects within the theory of convex geometry is that of the field of affine isoperimetric inequalities. Our focus deals with validating the upper bound of Petty's conjecture relating the volume of a convex body and that of its associated projection body. We begin our study by providing some background properties pertaining to convexity as seen through the lens of Minkowski theory. We then show that Petty's conjecture holds true in a certain class of 3-dimensional non-affine deformations of simplices. More precisely, we prove that any simplex in R^3 attains the upper bound in comparison to any deformation of a simplex by a Minkowski sum with a small line segment. As part of our theoretical analysis, we make use of mixed volumes and Maclaurin series expansion in order to simplify the targeted functionals. Finally, we provide an example validating what is known in the literature as the reverse and direct Petty projection inequality. In all cases, Mathematica is used extensively as our means of visualizing the plots of our selected convex bodies and corresponding projection bodies.