arXiv subjects

Antal Joós

Publications and source records attributed to Antal Joós.

5 recordsLinked to original sources

arXiv2023

Covering the Crosspolytope with Crosspolytopes

Let $γ^d_m(K)$ be the smallest positive number $λ$ such that the convex body $K$ can be covered by $m$ translates of $λK$. Let $K^d$ be the $d$-dimensional crosspolytope. It will be proved that $γ^d_m(K^d)=1$ for $1\le m< 2d$, $d\ge4$; $γ^d_m(K^d)=\frac{d-1}{d}$ for $m=2d,2d+1,2d+2$, $d\ge4$; $γ^d_m(K^d)=\frac{d-1}{d}$ for $ m= 2d+3$, $d=4,5$; $γ^d_m(K^d)=\frac{2d-3}{2d-1}$ for $ m= 2d+4$, $d=4$ and $γ^d_m(K^d)\le\frac{2d-3}{2d-1}$ for $ m= 2d+4$, $d\ge5$. Moreover the Hadwiger's covering conjecture is verified for the $d$-dimensional crosspolytope.

math.MG↗

arXiv2023

Isoperimetric problems for zonotopes

Shephard (Canad. J. Math. 26: 302-321, 1974) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note we prove a generalization of this theorem yielding similar formulas for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for d-dimensional zonotopes generated by d or d+1 segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the \ell_1$ polarization problem on the unit sphere and to a vector-valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.

math.MG↗

arXiv2022

Perfect packing of squares

It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={π^2/6}$. Meir and Moser asked what is the smallest $ε$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ can be packed into a rectangle of area ${π^2/6}+ε$. A packing into a rectangle of the right area is called perfect packing. Chalcraft packed the squares of sides of length $1$, $2^{-t}$, $3^{-t}$, $\ldots$ and he found perfect packing for $1/2<t\le3/5$. We will show based on an algorithm by Chalcraft that there are perfect packings if $1/2<t\le2/3$. Moreover we show that there is a perfect packing for all $t$ in the range $\log_32\le t\le2/3$.

math.CO↗

arXiv2022

Packing $1.35\cdot 10^{11}$ rectangles into a unit square

It is known that $\sum\limits_{i=1}^{\infty} \frac{1}{i (i+1)} = 1$. In 1968, Meir and Moser asked for finding the smallest $ε$ such that all the rectangles of sizes $1/i \times 1/(i + 1)$ for $i = 1, 2, \ldots$, can be packed into a unit square or a rectangle of area $1 + ε$. In this paper, we show that we can pack the first $1.35\cdot10^{11}$ rectangles into the unit square and give an estimate for $ε$ from this packing.

math.CO↗

arXiv2021

Cubes and Boxes have Rupert's passages in every direction

It is a $300$ year old counterintuitive observation of Prince Rupert of Rhine that in cube a straight tunnel can be cut, through which a second congruent cube can be passed. Hundred years later P. Nieuwland generalized Rupert's problem and asked for the largest aspect ratio so that a larger homothetic copy of the same body can be passed. We show that cubes and in fact all rectangular boxes have Rupert's passages in every direction, which is not parallel to the faces. In case of the cube it was assumed without proof that the solution of the Nieuwland's problem is a tunnel perpendicular to the largest square contained by the cube. We prove that this unwarranted assumption is correct not only for the cube, but also for all other rectangular boxes.

math.MG↗