129 research outputs found
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
Convex normality of rational polytopes with long edges
We introduce the property of convex normality of rational polytopes and give
a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing
the property. As an application, we show that if every edge of a lattice
d-polytope P has lattice length at least 4d(d+1) then P is normal. This answers
in the positive a question raised in 2007. If P is a lattice simplex whose
edges have lattice lengths at least d(d+1) then P is even covered by lattice
parallelepipeds. For the approach developed here, it is necessary to involve
rational polytopes even for applications to lattice polytopes.Comment: 16 pages, final version, to appear in Advances in Mathematic
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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