On the convex hull of uniform random points in a simpled-polytope
TLDR
The sharp constants of the asymptotic expansion ofEn(f), En(v), andEn(V), asn tends to infinity are determined.Abstract:
Denote the expected number of facets and vertices and the expected volume of the convex hullPn ofn random points, selected independently and uniformly from the interior of a simpled-polytope byEn(f), En(v), andEn(V), respectively. In this note we determine the sharp constants of the asymptotic expansion ofEn(f), En(v), andEn(V), asn tends to infinity. Further, some results concerning the expected shape ofPn are given.read more
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References
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An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
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An Introduction to Probability Theory and Its Applications.
Book
A Course of Modern Analysis
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Book
Integral geometry and geometric probability
Abstract: Part I. Integral Geometry in the Plane: 1. Convex sets in the plane 2. Sets of points and Poisson processes in the plane 3. Sets of lines in the plane 4. Pairs of points and pairs of lines 5. Sets of strips in the plane 6. The group of motions in the plane: kinematic density 7. Fundamental formulas of Poincare and Blaschke 8. Lattices of figures Part II. General Integral Geometry: 9. Differential forms and Lie groups 10. Density and measure in homogenous spaces 11. The affine groups 12. The group of motions in En Part III. Integral Geometry in En: 13. Convex sets in En 14. Linear subspaces, convex sets and compact manifolds 15. The kinematic density in En 16. Geometric and statistical applications: stereology Part IV. Integral Geometry in Spaces of Constant Curvature: 17. Noneuclidean integral geometry 18. Crofton's formulas and the kinematic fundamental formula in noneuclidean spaces 19. Integral geometry and foliated spaces: trends in integral geometry.