The symmetric plank problem, revisited
We present a streamlined proof of K. Ball's symmetric plank theorem in $\mathbb{R}^d$, which solves the affine plank problem raised by Th. Bang for symmetric convex bodies.
math.MG↗
arXiv subjects
Gergely Ambrus
Publications and source records attributed to Gergely Ambrus.
Uniform tight frames as optimal signals
Non-orthogonal communication is a promising technique for future wireless networks (e.g., 6G and Wi-Fi 7). In the vector channel model, designing efficient non-orthogonal communication schemes amounts to the following extremum problem: \[ \max \min_k \frac{|v_k|^2}{σ^2 + \sum_{l \neq k} \langle v_k, v_l \rangle^2} \] where the maximum is taken among vector systems $(v_k)_1^N \subset \mathbb{R}^d$ satisfying $c_1 \leq |v_k|^2 \leq c_2$ for every $k$, and the parameter $σ>0$ corresponds to the noise of the channel. We show that in the case $σ= 0$, uniform tight frames are the only optimal configurations. We also give quantitative bounds on the optimal capacity of vector channels with relatively small noise.
math.MG↗
New estimates for convex layer numbers
Starting with a finite point set $X \subset \mathbf{R}^d$, the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove $X$ is called the layer number of $X$, denoted by $L(X)$. In the article, we study the layer number of evenly distributed families of point sets contained in $B^d$, the $d$-dimensional unit ball. These sets consist of points in $B^d$ whose minimal distance is asymptotically as large as possible. We show that for a set $X$ belonging to an evenly distributed family, $L(X) \geq Ω(|X|^{1/d})$ holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that $L(X)\leq O(|X|^{2/d})$ holds for $d\geq 2$, which improves greatly on the current upper bound of $O(|X|^{(d+1)/2d})$ for $d \geq 3$. Finally, we provide a recursive construction of evenly distributed families whose sets satisfy $L(X) = Θ(|X|^{2/d - 1/(d 2^{d-1})})$, showing that our upper bound is nearly tight.
math.MG↗
Small subset sums
Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements such that \| \sum_{v \in U} v \| \le \lceil d/2 \rceil. We also prove that this bound is sharp in general. We improve the estimate to O(\sqrt d) for the Euclidean and the max norms. An application on vector sums in the plane is also given.
math.MG↗
Density estimates of 1-avoiding sets via higher order correlations
We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied.
math.MG↗
Longest k-monotone chains
We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild boundary conditions. Besides determining the order of magnitude of the expectation, we also prove strong concentration estimates. We provide a general framework that includes the previously studied cases of $k=1$ (longest increasing sequences) and $k=2$ (longest convex chains).
math.MG↗
The layer number of grids
The peeling process is defined as follows: starting with a finite point set $X \subset \mathbb{R}^d$, we repeatedly remove the set of vertices of the convex hull of the current set of points. The number of peeling steps needed to completely delete the set $X$ is called the layer number of $X$. In this paper, we study the layer number of the $d$-dimensional integer grid $[n]^d$. We prove that for every $d \geq 1$, the layer number of $[n]^d$ is at least $Ω\left(n^\frac{2d}{d+1}\right)$. On the other hand, we show that for every $d\geq 3$, it takes at most $O(n^{d - 9/11})$ steps to fully remove $[n]^d$. Our approach is based on an enhancement of the method used by Har-Peled and Lidický for solving the 2-dimensional case.
math.MG↗
Polarization, sign sequences and isotropic vector systems
We determine the order of magnitude of the $n$th $\ell_p$-polarization constant of the unit sphere $S^{d-1}$ for every $n,d \geq 1$ and $p>0$. For $p=2$, we prove that extremizers are isotropic vector sets, whereas for $p=1$, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for $d=2$, we discuss the optimality of equally spaced configurations on the unit circle.
math.MG↗
Analytic and Probabilistic Problems in Discrete Geometry
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence $u_1,\dots, u_n$ of norm 1 vectors in a real Hilbert space $\mathscr H$, there exists a unit vector $v \in \mathscr H$, such that $$ \sum \frac{1}{\langle u_i, v \rangle^2} \leq n^2. $$ The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take $n$ independent, uniform random points in a triangle $T$. Convex chains between two fixed vertices of $T$ are defined naturally. Let $L_n$ denote the maximal size of a convex chain. We prove that the expectation of $L_n$ is asymptotically $α\, n^{1/3}$, where $α$ is a constant between 1.5 and 3.5 -- we conjecture that the correct value is 3. We also prove strong concentration results for $L_n$, which, in turn, imply a limit shape result for the longest convex chains.
math.MG↗
A short proof for the characterisation of tight frames
With the aid of utilising tensor products, we give a simplified proof to the fundamental theorem of Benedetto and Fickus about the existence and characterisation of finite, normalised tight frames. We also establish unit-norm tensor resolutions for symmetric, positive semi-definite matrices.
math.CA↗
Linear programming duality for geometers
We present a geometric proof for the duality theorem of linear programming. Besides being self-contained and simple, the present approach also provides a transparent way for understanding this fundamental result.
math.MG↗
Extremal cross-polytopes and Gaussian vectors
Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised when C is at most two dimensional. As a corollary, a lower bound on the mean width of a general convex body K is derived in terms of the successive inner radii of K. A more general result is presented for Gaussian random vectors.
math.MG↗
Chebyshev constants for the unit circle
It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.
math.MG↗
The diminishing segment process
Let S(1) be the segment [-1,1], and define the segments S(n) recursively in the following manner: let S(n+1) be the intersection of S(n) and a(n+1) + S(1), where the point a(n+1) is chosen randomly on the segment S(n) with uniform distribution. For the radius r(n) of S(n) we prove that n(r(n) - 1//2) converges in distribution to an exponential law, and we also show that the centre of the limiting unit interval has arcsine distribution.
math.PR↗
Stability results for the volume of random simplices
It is known that for a convex body K in R^d of volume one, the expected volume of random simplices in K is minimised if K is an ellipsoid, and for d = 2, maximised if K is a triangle. Here we provide corresponding stability estimates.
math.MG↗
Longest convex chains
Assume $X_n$ is a random sample of $n$ uniform, independent points from a triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined naturally. The length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this article, we determine the order of magnitude of the expectation of $L_n$. We show further that $L_n$ is highly concentrated around its mean, and that the longest convex chains have a limit shape.
math.PR↗