arXiv subjects

D. Ryabogin

Publications and source records attributed to D. Ryabogin.

3 recordsLinked to original sources

arXiv2026

On convex bodies with constant non-central sections

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac 1{\pi} \arctan((\frac{3A}{4\pi})^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.

math.MG↗

arXiv2017

Generalized Grünbaum inequality

Let $f$ be an integrable log-concave function on ${\mathbb R^n}$ with the center of mass at the origin. We show that $\int\limits_0^{\infty}f(sθ)ds\ge e^{-n}\int\limits_{-\infty}^{\infty}f(sθ)ds$ for every $ θ\in S^{n-1}$, and the constant $e^{-n}$ is the best possible.

math.MG↗

arXiv2010

The behavior of iterations of the intersection body operator in a small neighborhood of the unit ball

The intersection body of a ball is again a ball. So, the unit ball $B_d \subset \R^d$ is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur distance.We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to $B_d$ in Banach-Mazur distance converge to $B_d$ in Banach-Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of $B_d$.

math.MG↗