We study the maximizers of the generalized volume product \[ \gamma_\sigma^n(A)\,\gamma_\sigma^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $\gamma_\sigma^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance $\sigma^2 I_n$, $\sigma>0$. It turns out that the maximizers depend on $\sigma$. We prove that they exist and are convex bodies. In dimension $n=1$, we find the exact form of the maximizers. In dimension $n\ge 2$, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for $\sigma^2 \le \frac{1}{{n}}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for $\sigma^2\ge {\frac{2}{n+1}}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for $\sigma^2 \le \frac{2}{3}$.
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schr\"odinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\f...
We prove a sharp Santal\'o-type inequality for the product of $V(\varphi)$ and $V$ of the Legendre transform of $\varphi$, where $\varphi$ is an even geometric convex function on $\mathbb{R}^n$ and $V(\varphi)=\int(1+\varphi)^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the s...
We study forward orbits in $\mathbb Z^2$ generated by the expanding affine maps $(x,y)\mapsto(mx+i,ny+j)$, where $m\ge n\ge2$ are integers and $(i,j)$ ranges over a nonempty digit set $\Lambda\subseteq\{0,\ldots,m-1\}\times\{0,\ldots,n-1\}$. We obtain explicit formulae for the mass, Beurling, discrete packing, and Asso...
We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\a...
Carlos Beltrán, J. Marzo, J. Ortega-Cerdà· 0 citations
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the eve...
Yin-Jie Li· 0 citations
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