55 research outputs found
An isomorphic version of the Busemann-Petty problem for arbitrary measures
We prove the following theorem. Let be a measure on with even
continuous density, and let be origin-symmetric convex bodies in so
that for any central hyperplane H. Then
We also prove this result with better constants
for some special classes of measures and bodies. Finally, we prove a version of
the hyperplane inequality for convex measures
Bezout Inequality for Mixed volumes
In this paper we consider the following analog of Bezout inequality for mixed
volumes: We show that the above
inequality is true when is an -dimensional simplex and are convex bodies in . We conjecture that if the above
inequality is true for all convex bodies , then must
be an -dimensional simplex. We prove that if the above inequality is true
for all convex bodies , then must be indecomposable
(i.e. cannot be written as the Minkowski sum of two convex bodies which are not
homothetic to ), which confirms the conjecture when is a
simple polytope and in the 2-dimensional case. Finally, we connect the
inequality to an inequality on the volume of orthogonal projections of convex
bodies as well as prove an isomorphic version of the inequality.Comment: 18 pages, 2 figures; an error in the isomorphic version of the
inequality is corrected (which improved the inequality
Wulff shapes and a characterization of simplices via a Bezout type inequality
Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii
we study the following Bezout type inequality for mixed volumes We show
that the above inequality characterizes simplices, i.e. if is a convex body
satisfying the inequality for all convex bodies , then must be an -dimensional simplex. The main idea of
the proof is to study perturbations given by Wulff shapes. In particular, we
prove a new theorem on differentiability of the support function of the Wulff
shape, which is of independent interest.
In addition, we study the Bezout inequality for mixed volumes introduced in
arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies
which is strictly larger than the set of all polytopes that are non-simplices.
We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly
indecomposable convex bodies
The geometry of p-convex intersection bodies
Busemann's theorem states that the intersection body of an origin-symmetric
convex body is also convex. In this paper we provide a version of Busemann's
theorem for p-convex bodies. We show that the intersection body of a p-convex
body is q-convex for certain q. Furthermore, we discuss the sharpness of the
previous result by constructing an appropriate example. This example is also
used to show that IK, the intersection body of K, can be much farther away from
the Euclidean ball than K. Finally, we extend these theorems to some general
measure spaces with log-concave and -concave measure
Characterization of Simplices via the Bezout Inequality for Mixed volumes
We consider the following Bezout inequality for mixed volumes:
It was shown previously that
the inequality is true for any -dimensional simplex and any convex
bodies in . It was conjectured that simplices
are the only convex bodies for which the inequality holds for arbitrary bodies
in . In this paper we prove that this is indeed
the case if we assume that is a convex polytope. Thus the Bezout
inequality characterizes simplices in the class of convex -polytopes. In
addition, we show that if a body satisfies the Bezout inequality for
all bodies then the boundary of cannot have strict
points. In particular, it cannot have points with positive Gaussian curvature.Comment: 8 page
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