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Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
Intersections Of Dilatates Of Convex Bodies, Stefano Campi, Richard J. Gardner, Paolo Gronchi
Intersections Of Dilatates Of Convex Bodies, Stefano Campi, Richard J. Gardner, Paolo Gronchi
Mathematics Faculty Publications
We initiate a systematic investigation into the nature of the function ∝K(L,ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK(L, ρ) that gives the (n - 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρ L) of ρL. The focus is on the concavity properties of αK (L, ρ). Of particular interest is the …
Gaussian Brunn-Minkowski Inequalities, Richard J. Gardner, Artem Zvavitch
Gaussian Brunn-Minkowski Inequalities, Richard J. Gardner, Artem Zvavitch
Mathematics Faculty Publications
A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality, the first of its type, is proved, together with precise equality conditions, and is shown to be the best possible from several points of view. A new Gaussian Brunn-Minkowski inequality is proposed and proved to be true in some significant special cases Throughout the study attention is paid to precise equality conditions and conditions on the coefficients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.
The Brunn-Minkowski Inequality, Richard J. Gardner
The Brunn-Minkowski Inequality, Richard J. Gardner
Mathematics Faculty Publications
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.
A Brunn-Minkowski Inequality For The Integer Lattice, Richard J. Gardner, Paolo Gronchi
A Brunn-Minkowski Inequality For The Integer Lattice, Richard J. Gardner, Paolo Gronchi
Mathematics Faculty Publications
A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the …