The Sprawl Conjecture for Convex Bodies

What is the average distance between two points on the sphere of radius n in a normed space? There is a natural choice of measure making this average-distance statistic into an affine invariant, and we explore the conjecture that the `2 and `∞ norms provide the two extreme values of this invariant on Rd for every d. 1. Introducing the sprawl Consider the following asymptotic geometric statistic on a metric space: take the average distance between two points on the sphere of radius r, measured relative to r, as r gets large. To make sense of averaging, one needs a measure. For a metric space (X, d) with basepoint x0 and measures μr on the spheres Sr = Sr(x0), let the sprawl of X be E(X) = E(X, d, x0, {μr}) := lim r→∞ 1 r ∫ Sr×Sr d(x, y) dμr(x) dμr(y), if the limit exists. Note that since 0 ≤ d(x, y) ≤ 2r, the value is always between 0 and 2. If E = 2, this means that one can typically pass through the origin when traveling between any two points on the sphere without taking a significant detour. (The name is intended to invoke urban sprawl: a higher value means a lack of significant shortcuts between points on the periphery of the “city.”) In locally finite graphs (and therefore in finitely generated groups with their Cayley graphs), we can just use the uniform measure (i.e., counting measure), since the sphere of radius n is a finite set. A second setting, and the focus of this note, is finite-dimensional normed spaces: take R with a choice of a convex, centrally symmetric polytope L. Then there is an intrinsically defined norm on R and there are natural probability measures on its metric spheres, defined below in §2—for instance, if L is the round circle in R, we recover Euclidean distance and measure proportional to arclength. Some first examples: E(R) = 1 with any norm (because the distance between points on the sphere is equally likely to be 0 or 2r). For the Euclidean norm on the plane, we do a simple integral to find that E(R, `) = 4 π ≈ 1.273, while for the sup-norm we get E(R , `∞) = 4 3 ≈ 1.333. The main conjecture would imply that all norms on the plane have sprawls pinched between these two values (see Figure 5). Sprawl was introduced in [6], and this note should be read as a close companion to that article. This asymptotic statistic is also discussed in [5, 4, 3] in the context of curvature conditions: negative curvature is associated with high sprawl. Indeed, outside of convex geometry, it is easy to find examples with maximal sprawl. In the hyperbolic plane with its natural measure, if two rays make an angle θ, then their Date: October 27, 2012. 1 2 DUCHIN LELIÈVRE MOONEY time–r points are at most 2r − c(θ) apart; this means that E(H) = 2. Similarly, in a regular tree of finite degree, E(T ) = 2. 1.1. Motivation and connections to other ideas. For us, the original question for general groups was suggested by ergodic techniques for studying statistical geometry in lattices, where asymptotic averages over large metric spheres are a key tool (see, for just one instance, [7, §3.3]). Similar geometric invariants have been introduced by other authors in quite different contexts. For instance, in a paper developing analytic techniques for general metric spaces [9], Yann Ollivier considers an L average-distance statistic on small balls (as opposed to our L average-distance on large spheres), which he calls the spread. Independently, and with motivations from category theory and biodiversity, Simon Willerton has defined his own spread in [10], which turns out to be a kind of L−1 asymptotic distance average. (The bibliography of that paper points to other metric notions from the same family of ideas.) 1.2. The Sprawl Conjecture. Below, we explore the values that the sprawl can take in finite-dimensional normed spaces in connection with the conjecture that the Euclidean norm and the sup norm achieve the extreme values among norms on R, for every d. There is a continuous one-parameter family of perimeters interpolating from the sphere to the cube, and the sprawl is continuous in L, so all of the values between those two endpoints are achieved as E(L) for some L. Conjecturally, this is everything. Sprawl Conjecture. The sprawl is an affine isoperimetric invariant. That is, {E(L) : perimeters L ⊂ R} = [ E(R, `), E(R, `∞) ] . If true, this would place the sprawl in good company among a large family of affine isoperimetric invariants of convex bodies, which are by definition geometric statistics whose extremes are achieved (in the centrally symmetric case) by cubes and round balls. (In §3, it will be explained that this conjecture also implies bounds on the sprawl of free abelian groups with any finite generating sets.) The current paper proves some results cited in [6] and gives an algorithm, some key calculations, and some empirical evidence for exploring this Sprawl Conjecture. One of the elements of interest in studying this conjecture is that it is (at least apparently) immune to the main tools for proving affine isoperimetric inequalities: rearrangements. For instance, it is known that the Mahler volume (defined in §6) is maximized by round balls; this is proved by applying a sequence of Steiner symmetrizations under which a convex body converges to a ball and using the fact that Mahler volume is monotonic under these transformations. Another example of an affine isoperimetric invariant is the average distance between points in a solid body rather than on its perimeter; here, one can apply the powerful BrascampLieb-Luttinger inequality to quickly deduce that a round ball realizes the minimum ([8], using [1]). Neither of these approaches yields results in our case—sprawl is not monotonic under any of the usual kinds of symmetrization, and it is not in the right form to apply any of the well-known analytic rearrangement inequalities—so we undertake experimental methods. 1.3. Acknowledgments. We thank the referees for careful reading and numerous helpful suggestions. THE SPRAWL CONJECTURE FOR CONVEX BODIES 3 2. Norms and measures induced by polytopes Definition 1. A convex body is a compact convex set in R with nonempty interior. We will use the word perimeter to mean the boundary of a centrally symmetric convex body in R. The induced norm on R for a perimeter L is the unique norm for which L is the unit sphere. (And these recover all possible norms on R, as L varies.) The cone  on a set A ⊂ R is {ta : 0 ≤ t ≤ 1, a ∈ A}. The cone measure μL on a perimeter is given by μL(A) = Vol(Â) Vol(L̂) for measurable subsets A ⊂ L. Induced norm d = 1/10