8,384 research outputs found
Optimal Area-Sensitive Bounds for Polytope Approximation
Approximating convex bodies is a fundamental question in geometry and has a
wide variety of applications. Given a convex body of diameter in
for fixed , the objective is to minimize the number of
vertices (alternatively, the number of facets) of an approximating polytope for
a given Hausdorff error . The best known uniform bound, due to
Dudley (1974), shows that facets suffice.
While this bound is optimal in the case of a Euclidean ball, it is far from
optimal for ``skinny'' convex bodies.
A natural way to characterize a convex object's skinniness is in terms of its
relationship to the Euclidean ball. Given a convex body , define its surface
diameter to be the diameter of a Euclidean ball of the same
surface area as . It follows from generalizations of the isoperimetric
inequality that .
We show that, under the assumption that the width of the body in any
direction is at least , it is possible to approximate a convex
body using facets. This bound is
never worse than the previous bound and may be significantly better for skinny
bodies. The bound is tight, in the sense that for any value of ,
there exist convex bodies that, up to constant factors, require this many
facets.
The improvement arises from a novel approach to sampling points on the
boundary of a convex body. We employ a classical concept from convexity, called
Macbeath regions. We demonstrate that Macbeath regions in and 's polar
behave much like polar pairs. We then apply known results on the Mahler volume
to bound their number
Existence and approximation of densities of chord length- and cross section area distributions
In various stereological problems an n-dimensional convex body is intersected with an (n−1)-dimensional
Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated
with the (n−1)-dimensional volume of such a random section is studied. This distribution is also known
as chord length distribution and cross section area distribution in the planar and spatial case respectively.
For various classes of convex bodies it is shown that these distribution functions are absolutely continuous
with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the
corresponding probability density functions
On the Combinatorial Complexity of Approximating Polytopes
Approximating convex bodies succinctly by convex polytopes is a fundamental
problem in discrete geometry. A convex body of diameter
is given in Euclidean -dimensional space, where is a constant. Given an
error parameter , the objective is to determine a polytope of
minimum combinatorial complexity whose Hausdorff distance from is at most
. By combinatorial complexity we mean the
total number of faces of all dimensions of the polytope. A well-known result by
Dudley implies that facets suffice, and a dual
result by Bronshteyn and Ivanov similarly bounds the number of vertices, but
neither result bounds the total combinatorial complexity. We show that there
exists an approximating polytope whose total combinatorial complexity is
, where conceals a
polylogarithmic factor in . This is a significant improvement
upon the best known bound, which is roughly .
Our result is based on a novel combination of both old and new ideas. First,
we employ Macbeath regions, a classical structure from the theory of convexity.
The construction of our approximating polytope employs a new stratified
placement of these regions. Second, in order to analyze the combinatorial
complexity of the approximating polytope, we present a tight analysis of a
width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering.
Finally, we use a deterministic adaptation of the witness-collector technique
(developed recently by Devillers et al.) in the context of our stratified
construction.Comment: In Proceedings of the 32nd International Symposium Computational
Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and
Computational Geometr
A Geometric Lower Bound Theorem
We resolve a conjecture of Kalai relating approximation theory of convex
bodies by simplicial polytopes to the face numbers and primitive Betti numbers
of these polytopes and their toric varieties. The proof uses higher notions of
chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds
on the g-numbers of the approximating polytopes are given, in terms of their
Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi
Computing Mixed Discriminants, Mixed Volumes, and Permanents
We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite matrices within a 2 O(n) factor. As a corollary, we show that the permanent of an nonnegative matrix and the mixed volume of n ellipsoids in R n can be computed within a 2 O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximated by an ellipsoid, the last algorithm can be used for approximating in polynomial time the mixed volume of n convex bodies in R n within a factor n O(n) .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42420/1/454-18-2-205_18n2p205.pd
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