On Brunn-Minkowski Type Inequalities And Related Compressive Sensing Problems

The researches of this thesis belong to the theory of compressive sensing and convex geometric analysis,they are closely related to the study of concentration inequalities,and they had been widely in the fields of image processing,informa-tion and analysis.This paper is devoted to the researches of the phase retrieval problem and Brunn-Minkowski type inequalities,which is one of the hot issues in the compressive sensing theory and Brunn-Minkowski theory.These topics in the dissertation refer to the phase retrieval problem,a problem of extreme value with respect to dual Lp-Brunn-Minkowski theory,Orlicz difference body and radial Blaschke-Minkowski homomorphisms.From combinatorial optimization to the physical science,phase retrieval is widely used in diffraction imaging,X-ray crystallography and electron mi-croscopy.Thus,we consider the phase retrieval problem,which is expressed as yi = ?<ai,x>|2,i ?1,2,…,m,where the date {yi} and the design vectors ai?Rn are known,the vector x ? Rn is unknown.In chapter 2,we introduce a concrete algorithm for solving the problem:Newton algorithm,which starting with a good initial guess by mean of truncated spectral method,and update it-eration by a Newton iteration step.We prove that this algorithm quadratically converges to the solution in the real case,and it is also true for the large residual problem.In chapter 3,we study the maximal p-dual surface area(0<p<n)of a given convex body under the volume preserving affine transformations.And we prove that,under assumption 1,the maximal p-dual surface area under the volume preserving affine transformation if and only if p-dual surface area measure is isotropic on Sn-1.Meanwhile,we establish some related results for p-dual surface isotropic convex body,moreover,we also obtain the isoperimetric ineaulity and its related properties.Estimating the volume of a special body(difference body or reflection body)in terms of the addition of convex bodies is very important.For example,the classical Rogers-Shephard inequality for the difference body.Motivated by the ideas of Hernandez Cifre and Yepes Nicolas in[64],we axe mainly interested in whether it is possible to bound from the ratio V(|K+?(-K)]*)/V(K*)by a constant c?,n>0 depending on the dimension and ?,not on the convex body?we give a negative answer to above question in chapter 4.We also show that the Brunn-Minkowski type inequality for polar bodies.Intersection bodies play an important role in the solution of the Busemann-Petty problem.Schuster in[97]introduced the notion of radial Blaschke-Minkowski homomorphisms,which is more general than the well known intersection body operator.With the development of Orlicz-Brunn-Minkowski theory,we establish the Brunn-Minkowski type inequality of the radial Blaschke-Minkowski homo-morphisms in the Orlicz setting.Finally,we also obtain the Brunn-Minkowski type inequality of a volume difference function in chapter 5.