Uniqueness theorems for non-symmetric convex bodies

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This dissertation involves the determination of convex bodies and the comparison of sections of convex bodies. Uniqueness of convex bodies via derivatives of section functions is first considered. The result generalizes work of Minkowski, Falconer, and Gardner. The next chapter involves an attempt to derive a non-symmetric analogue of the Busemann-Petty problem. A two-dimensional example of convex bodies K and L is given such that there are an infinite number of points for which the cross-sections of K are smaller than the corresponding cross-sections of L. However K has a larger area than L. The final chapter includes a proof that planar, twice-differentiable, strictly convex bodies are uniquely determined by the Gaussian covariogram. The Gaussian covariogram of a body K is a function that yields the standard Gaussian measure of the intersection of K with its translates.--From public.pdf.