arXiv subjects
Publications and source records attributed to Jacek Jakimiuk.
We prove stability estimates for the volume of central hyperplane projections of unit balls in $\ell_p^n(\mathbb{R})$, establishing the dual counterpart of the result of Chasapis, Nayar and Tkocz on stability of sections.
The class AC of integrands satisfying the atomic condition was introduced by De Philippis, De Rosa, and Ghiraldin in 2018. These integrands give rise to Almgren elliptic geometric functionals as proven by the second author and De Rosa in 2020. So far, it is not known how to verify the atomic condition for any particular integrand (apart from the area integrand and its class 2 neighbourhood) or how to construct, event artificial, members of AC. We reinterpret the atomic condition in terms of convex geometry shedding light on the structure of this class. We also propose quantitative versions of AC, different from the SAC and USAC defined by De Rosa and Tione, which we call the exposed condition EC and the quadratic exposed condition QEC. As is the case with USAC, the QEC is stable under class 2 perturbations and supports a Caccioppoli-type inequality; hence, is suitable for proving regularity of critical points. Moreover, we provide some conditions sufficient for EC in case the integrand is associated to a norm on the exterior power of $\mathbf{R}^{n}$. Finally, for all $k \ge 2$ and $n-k \ge 3$ we construct strictly polyconvex integrands -- associated to inner-product norms on $\bigwedge_{k} \mathbf{R}^{n}$ -- which fail the atomic condition, showing that strict polyconvexity is necessary but far from sufficient for AC.
For fixed $t_0 \in [0,\sqrt{3}]$ we give asymptotically sharp lower bounds on the quantity $L_K \text{vol}_{d-1}(K \cap H)$, where $H$ is a hyperplane at distance $t_0 L_K$ from the origin, $K$ is any symmetric isotropic convex body in $\mathbb{R}^d$, and $L_K$ stands for the isotropic constant of $K$.
We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).
We give a strengthening of the classical Khintchine inequality between the second and the $p$-th moment for $p \ge 3$ with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.
Eskenazis, Nayar and Tkocz have shown recently some resilience of Ball's celebrated cube slicing theorem, namely its analogue in $l^n_p$ for large $p$. We show that the complex analogue, i.e. resilience of the polydisc slicing theorem proven by Oleszkiewicz and Pelczyński, holds for large $p$ and small $n$, but does not hold for any $p > 2$ and large $n$.
We develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that geometric distribution minimises Rényi entropy of order infinity under fixed variance, among all discrete log-concave random variables in $\mathbb{Z}$. We also show that the quantity $\mathbb{P}(X=\mathbb{E} X)$ is maximised, among all ultra-log-concave random variables with fixed integral mean, for a Poisson distribution.