In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

/pdf/the-brunn-minkowski-inequality-3ympoqc7tt.pdf

The Brunn-Minkowski inequality

The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.

/pdf/lp-affine-isoperimetric-inequalities-122jjxd2dp.pdf

Lp Affine Isoperimetric Inequalities

Introduction Basic concepts Volume and the Fourier transform Intersection bodies The Busemann-Petty problem Intersection bodies and $L_p$-spaces Extremal sections of $\ell_q$-balls Projections and the Fourier transform Bibliography Index.

/pdf/fourier-analysis-in-convex-geometry-b1nm8e8938.pdf

Fourier Analysis in Convex Geometry

We prove that in E3 the Busemann-Petty problem, concerning cen- tral sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.

/pdf/a-positive-answer-to-the-busemann-petty-problem-in-three-1prwazgliu.pdf

A positive answer to the Busemann-Petty problem in three dimensions

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (ni 1)dimensional X-ray) gives the ((ni 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R n and leads to a unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative answer to the Busemann-Petty problem for n• 4 and the negative answer for n‚ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

Gaoyong Zhang

Papers

Intersection bodies and the Busemann-Petty inequalities in $\mathbb{R}^4$