2,237 research outputs found
The Beauty of Random Polytopes Inscribed in the 2-sphere
Consider a random set of points on the unit sphere in , which
can be either uniformly sampled or a Poisson point process. Its convex hull is
a random inscribed polytope, whose boundary approximates the sphere. We focus
on the case , for which there are elementary proofs and fascinating
formulas for metric properties. In particular, we study the fraction of acute
facets, the expected intrinsic volumes, the total edge length, and the distance
to a fixed point. Finally we generalize the results to the ellipsoid with
homeoid density.Comment: 18 pages, 4 figure
Intrinsic volumes of inscribed random polytopes in smooth convex bodies
Let be a dimensional convex body with a twice continuously
differentiable boundary and everywhere positive Gauss-Kronecker curvature.
Denote by the convex hull of points chosen randomly and independently
from according to the uniform distribution. Matching lower and upper bounds
are obtained for the orders of magnitude of the variances of the -th
intrinsic volumes of for . Furthermore,
strong laws of large numbers are proved for the intrinsic volumes of . The
essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman,
and the Efron-Stein jackknife inequality
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Polytopes from Subgraph Statistics
Polytopes from subgraph statistics are important in applications and
conjectures and theorems in extremal graph theory can be stated as properties
of them. We have studied them with a view towards applications by inscribing
large explicit polytopes and semi-algebraic sets when the facet descriptions
are intractable. The semi-algebraic sets called curvy zonotopes are introduced
and studied using graph limits. From both volume calculations and algebraic
descriptions we find several interesting conjectures.Comment: Full article, 21 pages, 8 figures. Minor expository update
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