Category Archives: Convex geometry

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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