When the sphere is not round I.

This is the first of a series of posts concerning the geometry of unit spheres in normed spaces with the aim of attracting the attention of fellow geometers to a number of delightful open problems.  I will start with a couple of problems posed by Juan Jorge Schäffer in his book ”Geometry of Spheres in Normed Spaces”, Lecture Notes in Pure and Appl. Math., vol. 20, Dekker, NY, 1976.

Let us define the girth of a normed space (X,\|\cdot\|) as the infimum of the lengths of all rectifiable, centrally-symmetric, closed curves on the unit sphere. This quantity can also be described in (at least) two other ways : (1) as twice the infimum of  the (intrinsic) distance between a pair of antipodal points on the unit sphere; (2) as twice the infimum of the lengths of non-contractible rectifiable curves on the projectivized space P(X) with the length metric induced from the unit sphere. This second description links the girth to the study of systoles in projective spaces endowed with Finsler metrics.

In his book, Schäffer made the following two conjectures about the girth of normed spaces—he does not assume the spaces are Banach:

Conjecture 1. The girth of a normed space is equal to the girth of its dual.

Conjecture 2. The girth of a normed space of dimension greater than two is at most 2\pi with equality if and only if the space is Euclidean.

Some years ago I was lucky enough to settle the first conjecture in the affirmative by using some  elementary considerations in symplectic geometry (the paper was published in the American Journal of Mathematics, 2006, vol. 128, no2, pp. 361-371, but you can also find it in the ArXiv). Therefore, it is really the second conjecture that I would like to talk about.

To understand why the conjecture is plausible we must recall a little gem of the theory of normed spaces: suppose you are a being living in a normed space of dimension two and you’ve heard somewhere that one of the most important constants in the universe is half the perimeter of the unit disc, so you go and measure it. What do you get? A Polish mathematician by the name of Golab did just that and he found that the value of “\pi” on a normed plane is between 3 and 4. Moreover, you only get three if your unit disc is linearly equivalent to a regular hexagon and you only get four if  your unit disc is a parallelogram. Notice furthermore that the girth of a normed plane is exactly the perimeter of its unit disc, so–in that case—having girth greater than 2\pi means ”squarish” (OK, this is not rigourous, but perhaps making it so is not worthwhile at this point).  Now take the unit ball of a normed space and consider its two-dimensional central sections (i.e., its intersections with two-dimensional subspaces). The perimeters of the unit discs in these sections are greater than or equal to the girth of the space. Therefore if the girth is greater than 2\pi, all the sections are “squarish”.

A vaguely stated principle in convex geometry is that of all the central sections of a symmetric convex body some are rounder (i.e. more elliptical) than others. In my view we (or at least I) are far  from understanding this principle, but it does find its way into some results such as

Dvoretzki’s theorem: Given a positive number \epsilon and a positive integer k, there exists a positive integer N(\epsilon,k) such that any normed space of dimension greater than N(\epsilon,k) has a k-dimensional subspace that is \epsilon-close—in the Banach-Mazur distance—to a Euclidean space.

In particular, the girth of an infinite-dimensional normed space is at most 2\pi. Schäffer remarks in his book that it is enough to prove the conjecture for three-dimensional normed spaces.

What puzzles me in approaching Schäffer’s conjecture is that I’m not sure whether the key to its solution is geometric or topological. When the sun is out and I feel optimistic, I start believing that the following question has an affirmative answer.

Question. Let \pi : E \rightarrow S^2 be a non-trivial two-dimensional vector bundle over the sphere. Assume that on every fiber E_x we have a norm \|\cdot\|_x varying continuously with the base point. Is it true that at some point z \in S^2 the perimeter of the unit disc in E_z is less than or equal to 2\pi ?



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Convex Bodies and Probability II

1. Introduction

In the previous post I described a few probability spaces associated to star and convex bodies. Now I’ll describe three basic classes of random variables defined on these spaces and how they lead to some interesting constructions in convex geometry.

2. Linear functionals as random variables

Given a probability measure \mu on V every linear functional \xi : V \rightarrow \mathbb{R} can be considered as a random variable and it makes sense to consider its distribution (i.e., the induced probability measure \xi_{\#}\mu on the real line) together with its moments

{\cal M}_p(\xi; \mu) := \int_{\mathbb{R}} t^p d(\xi_{\#}\mu)(t) = \int_V (\xi \cdot x)^p d\mu(x)  for (p \geq 0).

Other related random variables on V are \xi_+ := \max\{\xi, 0\} and |\xi|, which we shall also consider along with their moments {\cal M}_p (\xi_+ ; \mu) and {\cal M}_p (|\xi|; \mu) for p \geq 0 and, in fact, for any value of p for which we can make sense of the moments using the theory of Schwartz distributions. However, in this post we’ll just consider p \geq 0.

For the rest of this post, I’ll assume that all p-moments for these random variables exist whatever the value of p \geq 0.

3. Centroid bodies

Given a probability measure \mu on V its centroid is the unit vector c(\mu) \in V such that {\cal M}_1 (\xi; \mu) = \xi \cdot c(\mu) for every linear functional \xi. In other words

c(\mu) = \int_V x d\mu(x).

The centroid body

Given a probability measure \mu on V consider the function \xi \mapsto {\cal M}_1 (\xi_+ ; \mu). This function is homogeneous of degree one and, provided we make some very basic assumptions on the measure \mu, defines an asymmetric norm on V^*. The unit ball for this norm

\Gamma^*_1 (\mu) := \{\xi : {\cal M}_1 (\xi_+ ; \mu) \leq 1 \}

is the polar centroid body of the measure \mu. The dual body of \Gamma^*_1 (\mu), to be denoted as \Gamma_1 (\mu), is the centroid body of \mu.

Using a minimal amount of distribution theory (i.e., that the derivative of the function t \mapsto \max\{t,0\} is the Heaviside function), we can give an alternative description of the centroid body:

Notice that the differential of

\xi \mapsto {\cal M}_1 (\xi_+ ; \mu) = \int_V \max\{\xi \cdot x, 0\} d\mu(x)

is the vector-valued map

\xi \mapsto \int_V x H(\xi \cdot x) d\mu(x) =  \int_{\xi \cdot x \geq 0} x d\mu(x).

It follows that every point in the boundary of \Gamma_1 (\mu) is the integral of the vector-valued measure x \mu(x) over a half-space that contains the origin in its boundary.

The p-centroid body and its polar.

For every p > 0 we define the polar p-centroid body of the measure \mu as the set

\Gamma^*_p (\mu) := \{\xi \in V^* : {\cal M}_p (\xi_+ ; \mu) \leq 1\}

Under very general assumptions \Gamma^*_p (\mu) is a star body in V^* for every p > 0 and a convex body for every p \geq 1.

The p-centroid body of the measure \mu is the dual body of \Gamma^*_p (\mu) and will be denoted by \Gamma_p (\mu).

Remark. I’ll leave this post unfinished for a while, but here is the general idea of where things are going: (1) applying the constructions above to the different probability measures associated to convex bodies yields many of the interesting constructions in convex geometry; (2) it may be that these constructions are also interesting for rather general classes of measures (log-concave measures, for example); (3) there are deep and interesting inequalities related to these constructions; (4) there are other interesting constructions in probability and statistics (i.e. entropy, Fisher information, etc.) that have also been studied.

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Convex Bodies and Probability I

Version 1.2: I’ve added more detailed description of duality from a geometric, a mechanical, and an analytic viewpoint.

1. Introduction

This set of very informal notes on the geometry of convex bodies and its relation to the theory of probability is a personal attempt to simplify and organize for my own use a large portion of the theory of convex bodies. The guiding principle is that there are several measure or probability spaces that are naturally associated to convex bodies and that even well-known probabilistic constructions can yield surprising insights into the geometry of convex bodies.

In this first part my only aim is to list some natural constructions of probability spaces related to convex geometry.

2. Dramatis Personae

I’ll be working in a real vector space V of dimension n in which I’ll be careful not to impose any further structure such as a basis or an inner product.

The space of linear, real-valued functions on V (the dual vector space) will be denoted by V^*. Duality plays an important role in the theory. Physically speaking, passing between V and V^* (and we need some structure to do this) corresponds to passing between velocity and momentum. Recall that velocity is just kinematics, momentum is physics (you must take into account the properties of the medium and the moving object to convert velocity into momentum).

A star body is a compact set in V that contains a neighborhood of the origin and such that if a point x is in the body, the whole segment that joins x to the origin is contained in the body.

A convex body is a compact set in V that has non-empty interior and such that if x and y are in the body, then the whole segment that joins x and y is contained in the body.

Convex bodies that contains the origin in their interior are also a star bodies and so I’ll refer to them as star convex bodies.

Definition of the gauge function of a star body.
Given a star body S \subset V, its gauge function is the real-valued function on $V$ whose value at point v is the least non-negative number $lates t$ for which v belongs to the dilated set tS.

Note that when S is a convex body symmetric about the origin, then its gauge function is the norm whose unit ball is S.

The Grassmannian of k-planes in V—to be denoted by G_k(V)—is the set of all its k-dimensional linear subspaces. If we also consider an orientation in these subspaces, then I’ll talk of the Grassmannian of oriented k-planes, G_k^+(V). Mostly k will be either 1 or n-1 and for these values of k we can think of the Grassmannian of oriented k-planes as a sphere of dimension n-1.

3. Two basic constructions

A. Push-forward of measures, mass transfer, and induced probabilities

These three terms really represent the same idea: given a measure space X with measure \mu and a measurable map f from X to some measurable space Y, the push-forward of \mu by f is the measure whose value at some event U \subset Y is the measure of the pre-image event f^{-1}(U) in X. In probabilistic terms the push-forward measure is the probability induced on Y by the random variable f.

B. Duality

In contrast to the previous paragraph where we had three terms to denote one idea, here we only have one term to denote three (related) ideas. I’ll start with the most geometric and basic of the three:

Given a set S in a finite-dimensional vector space V, the dual of S is the set of all half-spaces that contain it.

Note that every half-space in V that contains the origin in its interior is uniquely given by a linear inequality of the form \xi \cdot x \leq 1, where \xi is a non-zero linear functional. In other words, the set of all half-spaces containing the origin their interior can be perfectly identified with the dual space V^* minus its origin.

Since star bodies, by definition, contain the origin in their interior, the set of half-spaces containing a star body S can be seen as a subset S^* \subset V^*. It is not hard to show that once we add the origin to S^*, which we will always do from now on, the resulting set is a star convex body.

It is not hard to see that the dual of S^*, now a subset of (V^*)^* = V, coincides with the convex hull of S. In the case S is a star convex body S(^*)^* = S, and this justifies the term “duality”.

The second idea comes from mechanics rather than geometry: if we recall that the usual definition of the kinetic energy of a particle of unit mass is one-half the square of the magnitude of its velocity, it makes perfect sense to define the kinetic energy in precisely this way when the magnitude of the velocity is measured by a (possibly asymmetric) norm associated to a convex body containing the origin in its interior

Definition of kinetic energy associated to a star
convex body K.
The kinetic energy of a particle of unit mass moving with velocity v is one-half the square of the gauge function of K evaluated at v.

Definition of the momentum of a particle.
Let K be a star convex body in V whose boundary is a smooth hypersurface and let L : V \rightarrow [0,\infty) be the kinetic energy associated to K. The momentum of a particle of unit mass moving with velocity v is the linear functional \xi : V \rightarrow \mathbb{R} whose value at a vector w is given by the derivative at t = 0 of the function t \mapsto L(v + tw). In other words the momentum of this particle is dL(v), the differential of L evaluated at v.

That looks a bit strange at first sight and it takes a while getting used to, but it’s the right thing to do. The map v \mapsto dL(v) taking V minus the origin to the dual space V^* (or taking velocity to momentum) is the Legendre transform.

The dual K^* in V^* of a star convex body K \subset V is its image under the Legendre transform. In other words, it is the set of momenta that correspond to velocities whose magnitudes do not exceed 1. This is exactly the same dual body we previously defined as the set of all half-spaces containing K.

The third idea comes for analysis and optimization: recall that a function f: V \rightarrow \mathbb{R} is said to be convex if

f((1-t) x + ty) \leq (1-t)f(x) + tf(y) (0 \leq t \leq 1).

Equivalently, f is convex if its epigraph

\{(x,s) \in V \times \mathbb{R}: f(x) \leq s \}

is a convex set in V \times \mathbb{R}. From a convex function f: V \rightarrow \mathbb{R} we can construct a convex function on f^* : V^* \rightarrow \mathbb{R} by setting

f^*(\xi) := \sup_{x \in V} \xi \cdot x - f(x).

The function f^* is usually called the conjugate function of f, but it is also called its Legendre transform, or its Legendre-Fenchel transform. As you may expect, (f^*)^* = f.

4. Probability spaces associated to star and convex bodies

Definition of the uniform probability measure.
Given a star or convex body S \subset V, we define the probability of an event U \subset S as the quotient of the volume of U by the volume of S. Volume is defined by any multiple of the Lebesgue (Haar) measure in V (the multiple is irrelevant since we are taking quotients).

The uniform probability measure is the simplest, most basic construction. Many other constructions are obtained by inducing (pushing forward) this probability onto other sample spaces by various natural maps.

Definition of the “solid-angle” measure.
Given a star body S \subset V we consider its boundary \partial S as sample space and define the probability of an event U \subset \partial S as the fraction of the volume of S contained in the solid cone formed by the union of all line segments joining the origin with the points of U.

In other words, the solid-angle measure is the probability on the boundary
of S induced from the uniform probability on S by the radial map S\setminus 0 \rightarrow \partial S.

For the next definition I will define the Gauss map as a map that takes a smooth (or smooth enough) oriented hypersurface M \subset V to the Grassmannian of oriented n-1 planes in V, G_{n-1}^+(V): a point x \in M is taken to the (n-1)-dimensional subspace that is parallel to the hyperplane tangent to M at x.

There is no need to use unit normal vectors and this usually complicates things because it introduces an Euclidean metric we don’t really need. I don’t want to go into a full explanation of the term “smooth enough”, but in some cases we may allow the Gauss map to be multiple-valued and this will not pose any problems. This happens, for example, when M is a convex hypersurface.

Definition of the cone-volume measure.
Given a convex body K \subset V containing the origin in its interior, we define a probability measure on the Grassmannian G_{n-1}^+(V) by first considering the solid angle measure on the boundary of K and then using the Gauss map to induce a probability measure on the Grassmannian.

It helps to see what this gives for a polytope. In this case the cone volume measure is atomic and each atom corresponds to a facet of the polytope. In fact, the atoms are the hyperplanes containing the facets of the polytope translated to the origin.

Here is a simple-looking problem that is still unsolved:

Open problem (E. Lutwak, D. Yang, and G. Zhang):
Given a probability measure on the Grassmannian G_{n-1}^+(V), when is it the cone-volume measure of a convex body?

When the convex body is symmetric about the origin and we can replace the Grassmannian of oriented planes by G_{n-1}(V) (also known as the dual projective space P(V^*)), this problem was solved only a couple of years ago: http://www.ams.org/journals/jams/2013-26-03/S0894-0347-2012-00741-3/

Definition of the dual uniform measure.
This will be the probability measure on K^* induced from the uniform probability measure on K by the Legendre transform.

Definition of the dual solid angle measure
This will be the probability measure induced on the boundary of K^* from the dual uniform measure by means of the radial map K^* \rightarrow \partial K^*. Alternatively, it is the measure induced from the solid angle measure by (the restriction of) the Legendre transform dL : \partial K \rightarrow \partial K^*.

Note that given any measure associated to a convex body containing the origin in its interior, it makes sense to look at the dual measure defined as its push-forward under the Legendre transform.

5. Measures with convexity properties

In Version 1.3 this will be short summary of the work of Prekopa and Borell on logarithmically concave measures, quasi-concave measures, and related notions. The idea being that it is not only interesting to work with measures associated to convex bodies, but that convex-geometric concepts arise naturally in probability and measure theory.

The Prékopa-Leindler inequality

This inequality is the workhorse of most of what follows and unfortunately it is usually stated in a strange way. However, although the inequality is really quite surprising, it can be presented very naturally as a reverse of Hölder’s inequalty that makes use of the geometry underlying the measure space (\mathbb{R}^n,dx).

To motivate the inequality recall that in (\mathbb{R}, dx) as in any other measure space we have the Cauchy-Schwarz inequality for real-valued square-integrable functions:

\int_{\mathbb{R}} f(x)g(x) \, dx \leq \left( \int_{\mathbb{R}} f(x)^2 \, dx\right)^{1/2} \left( \int_{\mathbb{R}} g(x)^2 \, dx \right)^{1/2}

However, (\mathbb{R}, dx) is a very particular measure space and we can dream that for some other “exotic” product f \star g we can have a reverse inequality

\int_{\mathbb{R}} f \star g (x) \, dx \geq \left( \int_{\mathbb{R}} f(x)^2 \, dx\right)^{1/2} \left( \int_{\mathbb{R}} g(x)^2 \, dx \right)^{1/2}

In its first and most basic version, the Prékopa-Leindler inequality says that this dream can become reality:

Given two bounded functions f,g : \mathbb{R}^n \rightarrow \mathbb{R}, define their sup-convolution

f \star g (t) := \sup \{f(x)g(y) : x + y = t \}.

Theorem (Prékopa). If f and g are two non-negative, square integrable functions on (\mathbb{R}, dx), then

\int_{\mathbb{R}} f \star g (x) \, dx \geq 2 \left( \int_{\mathbb{R}} f(x)^2 \, dx\right)^{1/2} \left( \int_{\mathbb{R}} g(x)^2 \, dx \right)^{1/2}.

This is already remarkable, but to me it is downright amazing that exactly the same product can be used to obtain reverse-versions of Hölder’s inequality in (\mathbb{R}^n,dx):

Theorem (Prékopa and Leindler). Let f and g be non-negative measurable functions on (\mathbb{R}^n, dx). If f^p is integrable and g^q is integrable for 1 < p < \infty and 1/p + 1/q = 1, then

\int_{\mathbb{R}^n} f \star g (x) \, dx \geq (p^n)^{1/p} (q^n)^{1/q} \left( \int_{\mathbb{R}^n} f(x)^p \, dx\right)^{1/p} \left( \int_{\mathbb{R}^n} g(x)^q \, dx \right)^{1/q}.

Recall that for these functions, Hölder’s inequality states that

\int_{\mathbb{R}^n} f(x) g(x) \, dx \leq \left( \int_{\mathbb{R}^n} f(x)^p \, dx\right)^{1/p} \left( \int_{\mathbb{R}^n} g(x)^q \, dx \right)^{1/q}.

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