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Arkadiy Aliev

Publications and source records attributed to Arkadiy Aliev.

4 recordsLinked to original sources
arXiv2025

Mahler-type volume inequality for convex bodies with tetrahedral symmetry

Let $ K $ be a convex body in $ \mathbb{R}^n $. We denote the volume of $ K $ by $ \vert K\vert $, and the polar body of its difference body $ K - K $ by $ (K - K)^{\circ} $. We provide a new proof of the well-known estimate \[ |K||(K - K)^{\circ}| \geq \frac{3}{2} \] for $ K \subset \mathbb{R}^2 $, with equality attained for a triangle. For $ K \subset \mathbb{R}^3 $ with tetrahedral symmetry, we prove that \[ |K| |(K - K)^{\circ}| \geq \frac{2}{3}, \] with equality attained for a tetrahedron.

math.MG↗
arXiv2023

The exact bound for the reverse isodiametric problem in 3-space

Let $K$ be a convex body in $\mathbb{R}^{3}$. We denote the volume of $K$ by $Vol(K)$ and the diameter of $K$ by $Diam(K).$ In this paper we prove that there exists a linear bijection $T:\mathbb{R}^{3}\to \mathbb{R}^{3}$ such that $Vol(TK)\geq \frac{\sqrt{2}}{12}Diam(TK)^3$ with equality if $K$ is a simplex, which was conjectured by Endre Makai Jr. As a corollary, we prove that any non-separable lattice of translates in $\mathbb{R}^{3}$ has density of at least $\frac{1}{12}$, which is a dual analog of Minkowski's fundamental theorem. Also we prove that $Vol(K)\geq \frac{1}{12}ω(K)^3$, where $K\subset \mathbb{R}^{3}$ is a convex body and $ω(K)$ is the lattice width of $K$. In addition, there exists a three-dimensional simplex $Δ\subset \mathbb{R}^3$ such that $Vol(Δ) = \frac{1}{12}ω(Δ)^3.$

math.MG↗
arXiv2022

New estimates for $d_{2,1}$ and $d_{3,2}$

Let $K$ be a convex body in $\mathbb{R}^{n}$. Let $ d_{n,n-1}(K)$ be the smallest possible density of a non-separable lattice of translates of $K$. In this paper we prove the estimate $d_{2,1}(K)\leq \frac{π\sqrt{3}}{8}$ for $K\subset \mathbb{R}^{2}$, with equality if and only if $K$ is an ellipse, which was conjectured by E. Makai. Also we prove the estimate $d_{3,2}(K)\leq\fracπ{4\sqrt{3}}$ for $K\subset\mathbb{R}^{3}$ using projection bodies.

math.MG↗