References for reasoning about the spectrum of a convex body?

Question

By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- what can I say about the spectrum of this matrix?

For example, when the convex body $B$ is just the unit ball by logic like this its covariance matrix is diagonal and all eigenvalues are the same. My intuition is that this is because the unit ball is equally "fat" in all dimensions.

However, while I can similarly guess at the spectrum of other shapes based on thinness or fatness, I'm not sure how to approach the question rigorously. Again falling back on intuition, I would expect that the closer $B$ is to a sphere (e.g. if it both contains a sphere of radius $r$ and is contained in a sphere of radius $r'$) the more similar its spectrum (roughly).

Does anyone know of references for this problem, or similar variants? Thanks.

Answer 1

A very direct result giving characteristic function control for uniform measures on compact convex sets and hence spectrum is [Kulikova& Prokhorov].

It sounds to me that all you want to study is the uniform distribution's spectrum on a convex body $K$. With some additional symmetries it is not surprising that [MO1] can lead to a strong claim, if you are to investigate this particular type of argument, then it is probably better to investigate uniform distributions on symmetric spaces instead of distributions on general convex bodies, like [Duenez].

For a general convex body, there is little thing to say. The study in this direction is mostly about an isotropic convex body. If your problem is invariant under affine transformations then it does not lose generality by assuming isotropic. The covariance matrix on an isotropic convex body can be studied by observing the behavior of inertia matrix [Aubrun1], seminal results due to R.Kannan， L.Lovasz and J.Bourgain exist. [Brazitikos et al.]

There are a series of works by S.G.Bobkov and his collaborators that studied the concentration of mass in isotropic convex bodies. They formalized what they called "Central limit property" in [Bobkov&Koldobsky] around p.50 Theorem 2. So in high dimension $d$ the behavior of convariance matrix is asymptotically known. See [Paouris] Theorem 1.6 also.[Bobkov&Nazarov] showed how the geometry of unconditional measures will look like. Later [Aubrun2] extends Bai-Yin theorem to yield a sharper bound on the inertia matrix of uniform sampling. (For more details about Bai-Yin theorem please see part 1 of this [MO2] post.)

I have heard that there is a combinatoric approach to the problem of distributions on a general convex body but never have a chance to look into it. The book I heard about is [Kolchin et.al]. (Not sure but perhaps helpful.)

If you are interested in the boundary of a convex body and a distribution over it(say a random matrix with entries of uniform on the sphere $S^n$)instead of the convex body itself, a correct reference to look into is [Mardia&Peter] Chap 10.

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Reference

[Kulikova& Prokhorov]Kulikova, Anna A., and Yu V. Prokhorov. "Uniform distributions on convex sets: Inequality for characteristic functions." Theory of Probability & Its Applications 47.4 (2003): 700-701.

[Duenez]Duenez, Eduardo. "Random matrix ensembles associated to compact symmetric spaces." Communications in mathematical physics 244.1 (2004): 29-61. https://arxiv.org/pdf/math-ph/0111005.pdf

[Aubrun1]Aubrun, Guillaume. "Sampling convex bodies: a random matrix approach." Proceedings of the American Mathematical Society 135.5 (2007): 1293-1303.

[Aubrun2]Aubrun, Guillaume. "Random Points in the Unit Ball of ℓ n p." Positivity 10.4 (2006): 755-759.

[Bobkov&Nazarov]Bobkov, Sergey G., and Fedor L. Nazarov. "On convex bodies and log-concave probability measures with unconditional basis." Geometric aspects of functional analysis. Springer Berlin Heidelberg, 2003. 53-69.

[Bobkov&Koldobsky] Bobkov, Sergey G., and Alexander Koldobsky. "On the central limit property of convex bodies." Geometric aspects of functional analysis. Springer Berlin Heidelberg, 2003. 44-52.

[Kolchin et.al]Kolchin, Valentin Fedorovich, Boris Aleksandrovich Sevastyanov, and Vladimir Pavlovich Chistyakov. "Random allocations." (1978).

[Paouris]Paouris, Grigoris. "Concentration of mass on convex bodies." Geometric and Functional Analysis 16.5 (2006): 1021-1049.

[Mardia&Peter]Mardia, Kanti V., and Peter E. Jupp. Directional statistics. Vol. 494. John Wiley & Sons, 2009.

[Brazitikos et al.]Brazitikos, Silouanos, et al. Geometry of isotropic convex bodies. Vol. 196. Providence: American Mathematical Society, 2014.