Random points in halfspheres

Question

1. What are the contributions mentioned in the paper "Random points in halfspheres∗" ?

2. What are the future works mentioned in the paper "Random points in halfspheres∗" ?

3. What is the new phenomenon not observed in the Euclidean case?

4. what is the spherical hull of the n?

Accepted Answer

This is what the authors investigate here, establishing the asymptotic behaviour, as n tends to infinity, of the expectation of several characteristics of Pn, such as facet and vertex number, volume and surface area.

Accepted Answer

It would be interesting to extend ( 33 ) to the remaining functionals Fi, i = 2,..., d−2.. It would be interesting to know whether the lower bound ( 32 ) can be improved.

Accepted Answer

The fact that the expectation E fd−1(Pn) has an explicit expression for each n and not only an asymptotic expression for n→∞, is one of the new phenomena not observed in the Euclidean case.

Accepted Answer

For each n ∈ N, X(n)1 , . . . , X (n) n are independent random points in S+e , each with distribution µ, and Pn is their spherical convex hull.

Accepted Answer

Using that f0(Pn) = f1(Pn) for d = 2 and f0(Pn) = f2(Pn)/2 + 2 a.s. for d = 3 (since Pn is almost surely simplicial and the Euler relation holds), the authors can obtain the explicit values of C(2) and C(3) by comparing (4) and (15).

Accepted Answer

If each set Ck contains one of the random points X (n) 1 , . . . , X (n) n , then δs(Pn, S+e ) ≤ α byLemma 8.1, thusP ( δs(Pn,S+e ) ≤ α ) ≥ P ({ X (n) 1 , . . . , X (n) n } ∩ Ck 6= ∅ for k = 1, . . . ,m ) .

Accepted Answer

Ever since the seminal articles by Rényi and Sulanke [14], [15], convex hulls of random points and their asymptotic behaviour when the number of points increases, have been a favourite topic in stochastic geometry.

Accepted Answer

The authors have { X (n) 1 , . . . , X (n) n } ∩ Ck 6= ∅ for k = 1, . . . ,m if and only if there are sets I1, . . . , Im+1 ⊂ {1, . . . , n} which partition {1, . . . , n} and are such that |Ii| ≥ 1 for i = 1, . . . ,m and X(n)r ∈ Ck for r ∈

Accepted Answer

The authors choose a measurable function g : [0,∞)→ R with∫ ∞ 0 g(r)rd dr = 1and define F : (Rd+1)d → R byF (r1u1, . . . , rdud) := g(r1) · · · g(rd)F (u1, . . . , ud), ri ≥ 0, ui ∈ Sd.Applying the linear Blaschke–Petkantschin formula ([16], Thm. 7.2.1) to F , the authors get∫ (Rd+1)d F dλd = ωd+1 2 ∫ G(d+1,d) ∫ Hd F∇d dλdd−1 ν(dH), (5)where λd−1 denotes the d-dimensional Lebesgue measure on H. Using polar coordinates to transform the integrals over (Rd+1)d and Hd, the authors obtain the statement of the lemma.

Accepted Answer

The expectation of the vertex number is related to that of the volume by the spherical counterpart of Efron’s identity, namely1− E f0(Pn+1) n+ 1 = 2 ωd+1 Eσ(Pn). (26)For the reader’s convenience, the authors recall the short proof.

Accepted Answer

The set of spherical polytopes is equipped with the topology induced by the spherical Hausdorff distance and the corresponding Borel structure.

Accepted Answer

The spherical Hausdorff distance of two nonempty compact sets K,M ⊂ Sd is defined byδs(K,M) := max { max x∈K min y∈M ds(x, y), max x∈M min y∈K ds(x, y) } .

Random points in halfspheres

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Random points in halfspheres∗" ?

2. What are the future works mentioned in the paper "Random points in halfspheres∗" ?

3. What is the new phenomenon not observed in the Euclidean case?

4. what is the spherical hull of the n?

5. What is the simplest way to obtain the values of C(2) and C(3)?

6. if a set contains one of the random points X (n) 1,?

7. What is the main topic of the paper?

8. if there are sets I1 and i1,...,?

9. What is the measurable function of the lemma?

10. what is the spherical counterpart of Efron’s identity?

11. What is the topology of a spherical polytope?

12. What is the spherical Hausdorff distance of two nonempty compact sets?

Citations

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Expected intrinsic volumes and facet numbers of random beta-polytopes

Beta polytopes and Poisson polyhedra: f-vectors and angles

Angles of random simplices and face numbers of random polytopes

EXPECTED f-VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF-SPHERE

References

Stochastic and Integral Geometry

über die konvexe Hülle von n zufällig gewählten Punkten

Random polytopes in smooth convex bodies

Stochastic Geometry, Spatial Statistics and Random Fields

Sharp concentration of random polytopes

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