arXiv subjects

E. Makai Jr.

Publications and source records attributed to E. Makai Jr..

3 recordsLinked to original sources

arXiv2021

On the number of antipodal or strictly antipodal pairs of points in finite subsets of $\mathbb{R}^d$, III

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position in ${\mathbb{R}}^d$, affinely spanning ${\mathbb{R}}^d$, is $n + d(d - 1)/2 - 1$. Let ${\underline{sa}}^s_d(n)$ be the minimum of the number of strictly antipodal pairs of points among any $n$ points in ${\mathbb{R}}^d$, with affine hull ${\mathbb{R}}^d$, and in strictly convex position. The value of ${\underline{sa}}^s_d(n)$ was known for $d \le 3$ and any $n$. Moreover, ${\underline{sa}}^s_d(n) = \lceil n/2\rceil $ was known for $n \ge 2d$ even, and $n \ge 4d+1$ odd. We show ${\underline{sa}}^s_d(n) = 2d$ for $2d+1 \le n \le 4d-1$ odd, we determine ${\underline{sa}}^s_d(n)$ for $d=4$ and any $n$, and prove ${\underline{sa}}^s_d(2d -1) = 3(d - 1)$. The cases $d \ge 5 $ and $d+2 \le n \le 2d - 2$ remain open, but we give a lower and an upper bound on ${\underline{sa}}^s_d(n)$ for them, which are of the same order of magnitude, namely $Θ\left( (d-k)d \right) $. We present a simple example of a strictly antipodal set in ${\mathbb{R}}^d$, of cardinality const\,$\cdot 1.5874...^d$. We give simple proofs of the following statements: if $n$ segments in ${\mathbb{R}}^3$ are pairwise antipodal, or strictly antipodal, then $n \le 4$, or $n \le 3$, respectively, and these are sharp. We describe also the cases of equality.

math.CO↗

arXiv2015

Volume product of planar polar convex bodies --- lower estimates with stability

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \subset {\mathbb R}^2$ is a convex body, then we have $|K| \cdot |[(K-K)/2)]^* | \ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if $K$ has $n$-fold rotational symmetry about $o$, then $|K|\cdot |K^*| \ge n^2 \sin ^2 ( π/n)$, with equality if and only if $K$ is a regular $n$-gon of centre $o$. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular $n$-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality $|K|\cdot |K^*| \ge n^2 \sin ^2 ( π/n)$ to bodies with $o \in {\text{int}}\,K$, which contain, and are contained in, two regular $n$-gons, the vertices of the contained $n$-gon being incident to the sides of the containing $n$-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

math.MG↗

arXiv2015

Centrally symmetric convex bodies and sections having maximal quermassintegrals

Let $d \ge 2$, and let $K \subset {\Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $ω\in S^{d-1}$, we have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $ω$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (\cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small $C^2$-perturbations, or $C^3$-perturbations of the Euclidean unit ball, respectively.

math.MG↗