Bounding marginal densities via affine isoperimetry (Q2827965)

Let \(\mu\) be a probability measure on \(\mathbb{R}^n\) with a bounded density \(f\) and let \(E\) be a \(k\)-dimensional subspace of \(\mathbb{R}^n\), then the density of the marginal \(\pi_E(\mu)\) on \(E\), for \(x \in E\), is given by NEWLINE\[NEWLINE f_{\pi_E(\mu)}(x) = \int_{E^{\perp}+x} f(y) \;dy. NEWLINE\]NEWLINE The authors investigate to what extent the marginal densities are bounded as well. Let \(\mu_{n,k}\) be the Haar probability measure on the Grassmann manifold \(G_{n,k}\). Then it is shown in Theorem 1.1 that for each \(1\leq k \leq n-1\), there exists \(\mathcal{A} \subseteq G_{n,k}\) with \(\mu_{n,k}(\mathcal{A}) \geq 1- e^{-kn}\) such that, for every \(E \in \mathcal{A}\), NEWLINE\[NEWLINE f_{\pi_E(\mu)}(x)^{1/k} \leq C \| f \|_{\infty}^{1/n} NEWLINE\]NEWLINE for all \(x \in E\), except possibly on a set of \(\pi_E(\mu)\)-measure less than \(e^{-kn}\). Interestingly, this probabilistic result is based on an affine-invariance property of particular integrals on the Grassmannian and affine Grassmannian. The proof of Theorem 1.1 uses new affinely invariant extremal inequalities for certain averages of \(f\) on \(G_{n,k}\) (Theorem 1.2).