The study of the geometry of convex bodies based on information about sections and
projections of these bodies has important applications in many areas of mathematics and …

We derive a formula connecting the derivatives of parallel section functions of an origin-
symmetric star body in Rn with the Fourier transform of powers of the radial function of the …

All GL (n) covariant star-body-valued valuations on convex polytopes are completely
classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out …

Associated with each body K in Euclidean n-space R\spn is an ellipsoid Γ\sb2K called the
Legendre ellipsoid of K. It can be defined as the unique ellipsoid centered at the body's …

Projection and intersection bodies define continuous and GL (n) contravariant valuations.
They played a critical role in the solution of the Shephard problem for projections of convex …

We consider several generalizations of the concept of an intersection body and show their
connections with the Fourier transform and embeddings in L p-spaces. These connections …

The Busemann-Petty problem asks whether symmetric convex bodies in ℝ n with smaller
(n− 1)-dimensional volume of central hyperplane sections necessarily have smaller n …

We study the asymptotic behavior, as the dimension goes to infinity, of the volume of
sections of the unit balls of the spaces ℓ qn, 0< q≤∞. We compute the precise asymptotics …

We introduce complex intersection bodies and show that their properties and applications
are similar to those of their real counterparts. In particular, we generalize Busemann's …

This chapter belongs to the area of geometric tomography, which is the study of geometric
properties of solids based on data about their sections and projections. We describe a new …