Open Access. Powered by Scholars. Published by Universities.®

Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

1999

Western Washington University

Articles 1 - 3 of 3

Full-Text Articles in Mathematics

P-Cross-Section Bodies, Richard J. Gardner, Apostolos Giannopoulos Jul 1999

P-Cross-Section Bodies, Richard J. Gardner, Apostolos Giannopoulos

Mathematics Faculty Publications

If K is a convex body in En, its cross-section body CK has a radial function in any direction u is ∈ Sn-1 equal to the maximal volume of hyperplane sections of K orthogonal to u. A generalization called the p-cross-section body CpK of K, where p > -1, is introduced. The radial function of CpK in any direction u ∈ Sn-1 is the pth mean of the volumes of hyperplane sections of K orthogonal to u through points in K. It is shown that C …

Go to article

An Analytic Solution To The Busemann-Petty Problem On Sections Of Convex Bodies, Richard J. Gardner, Alexander Koldobsky, Thomas Schlumprecht Mar 1999

An Analytic Solution To The Busemann-Petty Problem On Sections Of Convex Bodies, Richard J. Gardner, Alexander Koldobsky, Thomas Schlumprecht

Mathematics Faculty Publications

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 body. A parallel section function (or (n - 1)-dimensional X-ray) gives the ((n - 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 Rn and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in Rn such that the …

Go to article

Extremal Graphs For Weights, Béla Bollobás, Paul Erdös, Amites Sarkar Jan 1999

Extremal Graphs For Weights, Béla Bollobás, Paul Erdös, Amites Sarkar

Mathematics Faculty Publications

Given a graph G = (V,E) and α ∈ R, we write wα(G)=∑xyϵEdG(x)αdG(y)α, and study the function wα(m) = max {wα(G): e(G) = m}. Answering a question from Bollobás and Erdös (Graphs of external weights, to appear), we determine wi(m) for every m, and we also give bounds for the case α ≠ 1.

Go to article