On Wendel’s equality for intersections of balls

We study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}$$\end{document} that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for n≥d+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge d+1$$\end{document} points in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}$$\end{document}, for the spindle convex hull this happens for all n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}. We study this question for the uniform and normally distributed random models.


Introduction and results
Wendel's equality [10] is one of the classical results in geometric probability: it states that if x 1 , . . . , x n are i.i.d. random points in R d whose distribution is (centrally) symmetric with respect to the origin o, and the probability measures of hyperplanes are 0, then the probability that o is not contained in the convex hull [x 1 , . . . , x n ] is One can find a simple proof of (1.1) in Bárány [1, pp. 94-95], which is independent of the distribution (under the above conditions). It was proved by Wagner and Welzl [9], that o-symmetric distributions are extremal in this sense. For more information, see also [8,Section 8.1.2].
Recently, Kabluchko and Zaporozhets [3] investigated the related problem of finding the probability that the convex hull of n i.i.d. normally distributed random points in R d contains a fixed points of space; they called these absorption probabilities. For a general introduction to random polytopes we refer to the recent survey paper by Schneider [7] and the book by Schneider and Weil [8].
We denote the d-dimensional origin centered unit radius closed ball by B d and its boundary by S d−1 . The symbol κ d denotes the volume (Lebesgue measure) of B d , and ω d is the surface volume of B d . For general information on convex sets, see the monograph [6] by Schneider. In this paper we study the following spindle convex variant of the above problems. Let x, y ∈ R d be two points and > 0. If |x − y| ≤ 2 , then let the spindle [x, y] determined by x and y be the intersection of all radius closed balls that contain both x and y. If |x − y| > 2 , then let [x, y] = R d . We say that a convex body K ⊂ R d (compact convex set with non-empty interior) is spindle convex with radius , or -spindle convex if together with any two points x, y ∈ K, it contains the spindle [x, y] . It is known ( [2]) that if a convex body K ⊂ R d is spindle convex with radius , then K is the intersection of all radius closed balls that contain K. This latter property is called radius ball-convexity.
Let First, we describe the -spindle convex uniform model. Let > 0, and let K ⊂ R d be an o-symmetric convex body that is -spindle convex. Let x 1 , . . . , x n be i.i.d. uniform random points from K. We denote the radius spindle convex hull of x 1 , . . . , x n by K (n) = [x 1 , . . . , x n ] . By the -spindle convexity of K, the random ball-polytope K (n) is contained in K. We ask the same question as in the classical convex case: what is the probability that o ∈ K (n) ? We note that in this model we may always achieve by scaling (simultaneously K and radius circles) that = 1. Henceforth, in the following two theorems we assume that = 1.
We study the special case when K = rB d with 0 < r ≤ 1. Then K is clearly spindle convex with radius = 1. We wish to determine the probability P (d, r, n) := P(o ∈ [x 1 , . . . , x n ] 1 ).
Finally, in Sect. 4, we study the Gaussian -spindle convex model. Let x 1 , . . . , x n be i.i.d. random points from R d distributed according to the standard normal distribution. The question is the same, what is the probability that o ∈ K (n) ? We note that in this second case, it may, and does, happen that K (n) = R d . We give an integral formula for the probability that a Gaussian unit radius spindle contains the origin and evaluate it numerically in the plane.

Proof of Theorem 1.1
Note that it is the simplest case of the model when n = 2, and K = rB d , where 0 < r ≤ 1 is a fixed number. This, of course, is of no interest in the classical version of Wendel's problem as Let us examine what it means geometrically that o ∈ [x 1 , x 2 ] 1 . Let M (x 1 ) denote the union of all open unit balls that contain o and x 1 on their boundary. Let K(d, r, x 1 ) be the part of rB d \M (x 1 ) that is in the closed half-space bounded by the hyperplane through o and orthogonal to x 1 which does not contain x 1 . We depicted this region in Fig. 1 when d = 2. We will only use K(2, r, x 1 ) in our calculations, so, in order to simplify notation, we will denote it by K(r, x 1 ) = K(2, r, x 1 ).
In order to evaluate P (d, r, 2), we use the linear Blaschke-Petkantschin formula. Let G(d, 2) denote the Grassmannian manifold of 2-dimensional linear subspaces of R d , and ν 2 be the unique rotation invariant Haar probability measure on G(d, 2). The 2-dimensional special case of the linear Blaschke-Petkantschin formula (see, for example, [8, Theorem 7.2.1 on p. 271]) says the following: If f : (R d ) 2 → R is a non-negative measurable function, then where ∇ 2 denotes the area of the parallelogram spanned by the vectors x 1 , x 2 in L. The symbol λ denotes the Lebesgue measure in R d , and λ L the (2dimensional) Lebesgue measure in L. Next, using polar coordinates for x 1 , x 2 ∈ L, that is, x 1 = r 1 u 1 , x 2 = r 2 u 2 , where u 1 , u 2 ∈ S 1 , r 1 , r 2 ∈ R + , we may write the right-hand-side of (2.1) as follows.
Vol. 97 (2023) On Wendel's equality for intersections of balls 443 By the rotational symmetry of rB d , integration with respect to u 1 is a multiplication by 2π. Hence, from now on, we fix u 1 = (0, 1). Let ϕ be the angle of u 2 and −u 1 , as shown on Fig. 1, and let ϕ(r 1 , r 2 ) = arcsin(r 1 /2) + arcsin(r 2 /2). Then The above integral can be evaluated for any specific value of d using multiple integration by parts. In particular,  Proof. Note that, using arcsin x ≤ πx/2 for x ∈ [0, π/2] and sin x ≤ x for x ∈ [0, π/2], we get that where the constant C(d) depends only on the dimension d. From this it follows that lim r→0 + P (d, r, 2) = 0 for d ≥ 2, as claimed.
In the proof of the second statement we use the fact that ϕ(r 1 , r 2 ) ≤ π/3. Thus

Proof of Theorem 1.2
The case when n = 3, can be treated, at least in the plane, as follows. We only consider when r = 1, that is, K = B 2 . Let x 1 , x 2 , x 3 be i.i.d. uniform random points from B 2 . Let Due to the rotational symmetry of B 2 , we may assume that x 1 = (0, r 1 ). Let x 2 = r 2 u 2 , where ϕ is the angle of u 2 and the negative half of the y-axis. Making use of the previously introduced notation, we write K(x 1 ) = K(1, x 1 ) and, similarly, K(x 2 ) = K(1, x 2 ). The ray ox i divides K(x i ) into two congruent parts. The part that is on the positive side of ox i is denoted by K + (x i ), and the negative part is K − (x i ), as shown in Fig. 2.
Finally, we note that according to Wendel's equality (1.1),

The case of normally distributed random points
In this subsection we consider the model in which = 1 and x 1 , . . . , x n are i.i.d. random points in R d that are distributed according to the standard normal distribution with density function Here we need to use the part of the definition of the spindle convex hull that normally does not come into play when the random points are chosen from a convex body that is spindle convex with radius less than or equal to 1. Namely, if x, y ∈ R d are such that |x − y| > 2, then [x, y] 1 := R d .
We are interested in the following probability It is clear that  [5, p. 438] and the historical references therein) that the density of s := l 2 /4 is Thus, where Γ(·) is Euler's gamma function, and γ(d/2, x) denotes the lower incomplete gamma function.
By the rotational symmetry of the normal distribution, integration with respect to u 1 is just a multiplication by 2π. Then, w obtain that The above integrals can be evaluated, at least numerically, for any specific value of d. In particular, for d = 2, we obtain for the first integral Data availibility Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
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