Sharp inequalities for the mean distance of random points in convex bodies☆
Introduction
One of the most classical questions in the area of geometric probability is Sylvester's question [30], which asks for the probability that the convex hull of four independently and uniformly distributed random points , in a planar compact convex set is a triangle. For particular sets K the precise value of is known and we refer to [17, Sections 2.31–2.34], [25] and [29, Chapter 5] for an extensive discussion. We also collect some examples in Table 1. Using symmetrization arguments, Blaschke [5] was able to prove that for any compact convex set with non-empty interior the two-sided inequality holds. A glance at Table 1 shows that the lower bound is achieved if (and, in fact, only if) K is an ellipse, and the upper bound if (and, in fact, only if) K is a triangle. In this context one should note that is invariant under affine transformations in the plane, which implies that the precise form of the ellipse and triangle does not play a role. It is not hard to verify that where stands for the area of K, see [27, Equation (8.11)]. Therefore, Blaschke's inequality (1) is equivalent to which gives the optimal lower and upper bound for the normalized mean area of the random triangle with vertices uniformly distributed in a planar compact convex set.
In the present paper we take up this classical and celebrated topic and instead of three points consider the situation where only two random points uniformly distributed in a compact convex set with non-empty interior are selected. In this case, their convex hull is a random segment having a random length. It is thus natural to ask for the optimal bounds of the normalized average length of this segment. While the area of the random triangle is normalized by the area of K, the length of the random segment should be normalized by the perimeter of K denoted by . In this paper we will prove that for any compact convex set with non-empty interior the inequality holds, where denotes the Euclidean distance of and . We emphasize that in contrast to (2) the inequalities on both sides of (3) are strict, and we shall argue that (3) is in fact optimal. Moreover, it will turn out that both bounds cannot be achieved by planar compact convex sets with interior points. In fact, the extremal cases correspond to two different degenerate situations, which will be described in detail. We would like to stress at this point that this surprising degeneracy phenomenon has only rarely been observed in similar situations so far in the existing literature around convex geometric inequalities. As such an exception we mention the inequalities for angle sums of convex polytopes by Perles and Shephard [24].
Remarkably, we will be able to derive the analogue of (3) in any dimension , where instead of the perimeter one normalizes the mean distance by the so-called first intrinsic volume of K, which in turn is a constant multiple of the mean width. We emphasize that this is in sharp contrast to Blaschke's inequality (2) for which only a lower bound is known in any space dimension. This is the context of Busemann's random simplex inequality for which we refer to [9] or [27, Theorem 8.6.1] (according to results of Groemer [14], [15] this holds more generally for convex hulls generated by an arbitrary number of random points and also for higher moments of the volume). A corresponding upper bound is still unknown, but in view of the planar case, it seems natural to expect that a sharp upper bound is provided by d-dimensional simplices. This is known as the simplex conjecture in convex geometric analysis and a positive solution would imply the famous hyperplane conjecture, see [23] or [7, Corollary 3.5.8].
The remaining parts of this paper are structured as follows. In Section 2 we start with some historical remarks of what is known about the so-called mean distance of convex bodies. Our main result is presented in Section 3. Its proof is divided into several parts: proof of the lower bound (Section 4.2), proof of the upper bound (Sections 4.3–4.5) and sharpness of the estimates (Section 4.6).
Section snippets
Historical remarks
Before presenting our main results, we start with some historical remarks, which should help reader to bring our results in line with what is known from the literature. We also introduce some basic notation that will be used throughout the paper.
By a convex body in we understand a compact convex subset of with non-empty interior. Let be a convex body and let and be two independent random vectors uniformly distributed in K. We will denote by the mean distance between and
Main result
Let be a convex body. The main goal of this paper is to derive the optimal lower and upper bounds for normalized by the mean width of K, which is given by where denotes the length of the projection of K onto the line spanned by u.
An obstacle when working with the mean width is its dependence on the dimension of the ambient space. In fact, if we embed K into with , then is strictly decreasing with respect to n. That is why it is convenient to use
Preliminaries
Before presenting the proof of Theorem 1 we start with some general comments on . It follows from (9) and (8) along with Fubini's theorem that Let us fix some . Again by Fubini's theorem, we see that where Let be an affine function which maps the interval
Acknowledgement
This project has been initiated when DZ was visiting Ruhr University Bochum in September and October 2019. Financial support of the German Research Foundation (DFG) via Research Training Group RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity is gratefully acknowledged. We also thank an anonymous referee for insightful comments and remarks which helped us to further improve our paper. We also thank Uwe Bäsel for pointing us to the correct value for in
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The work of GB and AG was partially supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-Dimensional Phenomena in Probability-Fluctuations and Discontinuity. The work of DZ was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” and by RFBR and DFG according to the research project 20-51-12004.