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on Isotropic constant of a log-concave probability measure ($C_{20b}$)

The statement

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ with finite second moments. Its covariance matrix is

\mathrm{Cov}(\mu) \ :=\ \int_{\mathbb{R}^n} (x-m)(x-m)^{\mathsf T}\, d\mu(x), \qquad m:=\int_{\mathbb{R}^n} x\, d\mu(x).

Convex bodies

If $K\subset\mathbb{R}^n$ is a convex body, let $\lambda_{K}$ be the uniform probability measure on $K$ and abbreviate $\mathrm{Cov}(K):=\mathrm{Cov}(\lambda_{K})$ . The isotropic constant of $K$ is

L_{K} \ :=\ \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)}.

This quantity is invariant under invertible affine maps.

Define

L_{n}^{\mathrm{body}} \ :=\ \sup\{L_{K}:\ K\subset\mathbb{R}^n \text{ a convex body}\}.

Log-concave probability measures

If $\mu$ is absolutely continuous with density $f$ , its (differential) entropy is

\mathrm{Ent}(\mu)\ :=\ -\int_{\mathbb{R}^n} f \log f.

For an absolutely-continuous log-concave probability measure $\mu$ on $\mathbb{R}^n$ , define its isotropic constant by

L_\mu\ :=\ e^{-\mathrm{Ent}(\mu)/n}\cdot \det(\mathrm{Cov}(\mu))^{1/(2n)}.

(If $\mu$ is log-concave but supported on a proper affine subspace, define $L_{\mu}$ in that subspace.)

If $K$ is a convex body, then $\lambda_{K}$ has constant density $1/\mathrm{Vol}_{n}(K)$ on $K$ , hence $\mathrm{Ent}(\lambda_{K})=\log\mathrm{Vol}_{n}(K)$ , and therefore

L_{\lambda_{K}} := e^{-\log(\mathrm{Vol}_{n}(K))/n}\cdot \det(\mathrm{Cov}(K))^{1/(2n)} := \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)} := L_{K}.

Define

L_{n}^{\mathrm{lc}} \ :=\ \sup\{L_\mu:\ \mu \text{ log-concave on }\mathbb{R}^n\}, \qquad C_{20b} \ :=\ \sup_{n\ge 1} L_{n}^{\mathrm{lc}}.

The isotropic constant problem asked whether $C_{20b}<\infty$ (i.e. whether $L_\mu$ is bounded by a universal constant, independent of the dimension). This is now known to be true.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.