CyberSec Research

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in $\mathbb R^3$: for $n \ge 2$ let $p_1, \ldots,p_n \in [0,1]^3$ be a collection of points and let $\ell_i$ be a line through $p_i$ for every $i$ such that $d(p_i, \ell_j) \ge \delta$ for all $i\neq j$. Then we have $n \lesssim \delta^{-3+\gamma}$ for some absolute constant $\gamma>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.

The intersection body of order $i$ for $i=1,2,\ldots,n-2$, $I_iK$, of a star body $K$ in $\mathbb{R}^n$ introduced by G. Zhang, plays a central role in the dual Brunn-Minkowski theory. We show that when $n \geq 3$, $I_i^2K = cK$ iff $K$ is an origin-symmetric ball, and hence $I_iK = cK$ iff $K$ is an origin-symmetric ball. Combining the breakthrough (case $i = n-1$) of Milman, Shabelman and Yehudayoff (Invent. Math., 241 (2025), 509-558), two long-standing questions 8.6 and 8.7 posed by R. Gardner (Page 302, Geometric Tomography, Cambridge University Press, 1995) are completely resolved. An equivalent formulation of the latter in terms of non-linear harmonic analysis states that a non-negative $\rho\in L^{\infty}(\mathbb{S}^{n-1})$ satisfies $\mathcal{R}(\rho^i) = c\rho$ for some $c > 0$ iff $\rho$ is constant, where $\mathcal{R}$ is the spherical Radon transform. As applications, the generalized Busemann intersection inequalities are established.

For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we establish the following two results for an $n$-point metric space $X$ of negative type : $\lim_{t\to 0}{\rm Mag}\ Z_{tX} = 1$ and ${\rm Mag}\ Z_{X}< n$ for $n>1$. The second result gives a negative answer to the problem given by Gomi--Meckes. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product.

Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an external point is called a cap. The union of finitely many caps of a ball is a cap body if it is a convex set. In this note, we confirm the Hadwiger's conjecture for the class of cap bodies in all dimensions, bridging recently established cases of $n=3$ and large $n$. For $4\le n\le 15$, the proof combines a probabilistic technique with reduction to linear programming performed with computer assistance. For $n\ge 9$, we give an explicit bound on illumination number of cap bodies based on the same probabilistic technique but avoiding computer aid.

The attractors of iterated function systems are usually obtained as the Hausdorff limit of any non-empty compact subset under iteration. In this note we show that an iterated function system on a boundedly compact metric space has compact, invariant subspaces so that the attractor of the IFS can also be expressed as the intersection of a sequence of decreasing compact spaces.

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are representable by integration with respect to the normal cycle. This allows us to reduce the statement to a corresponding result for the representation of $\mathfrak{sl}(n)$ on the space of these differential forms.

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few and Rogers [Mathematika 6, 1959] and the upper bound of $\left(1/2+o(1) \right)n \ln n$ by Dumer [Discrete Comput. Geom. 38, 2007]. We prove that there are ball coverings of $\mathbb{R}^n$ attaining the asymptotically best known density $\left(1/2+o(1) \right)n \ln n$ such that, additionally, every point of $\mathbb{R}^n$ is covered at most $\left(1.79556... + o(1)\right) n \ln n$ times. This strengthens the result of Erd\H{o}s and Rogers [Acta Arith. 7, 1961/62] who had the maximum multiplicity at most $\left(\mathrm{e} + o(1)\right) n \ln n$. On the other hand, we show that the method that was used for the best known ball coverings (when one takes a random subset of centres in a fundamental domain of a suitable lattice in $\mathbb{R}^n$ and extends this periodically) fails to work if the density is less than $(1/2+o(1))n\ln n$; in fact, this result remains true if we replace the ball by any convex body $K$. Also, we observe that a ``worst'' convex body $K$ here is a cube, for which the packing density coming from random constructions is only $(1+o(1))n\ln n$.

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves.

We study point configurations on the torus $\mathbb T^d$ that minimize interaction energies with tensor product structure which arise naturally in the context of discrepancy theory and quasi-Monte Carlo integration. Permutation sets on $\mathbb T^2$ and Latin hypercube sets in higher dimensions (i.e. sets whose projections onto coordinate axes are equispaced points) are natural candidates to be energy minimizers. We show that such point configurations that have only one distance in the vector sense minimize the energy for a wide range of potentials, in other words, such sets satisfy a tensor product version of universal optimality. This applies, in particular, to three- and five-point Fibonacci lattices. We also characterize all lattices with this property and exhibit some non-lattice sets of this type. In addition, we obtain several further structural results about global and local minimizers of tensor product energies.

It is well known that if a metric space is uniformly disconnected, then its conformal dimension is zero. First, we characterize when a self-affine sponge of Lalley-Gatzouras type is uniformly disconnected. Thanks to this characterization, we show that a self-affine sponge of Lalley-Gatzouras type has conformal dimension zero if and only if it is uniformly disconnected.

We study multiplicative inequalities among entries of Lorentzian matrices, referred to as bounded ratios. These inequalities can be viewed as generalizations of the classical Alexandrov--Fenchel inequalities for mixed volumes. Our main structural result identifies the cone of all bounded ratios on Lorentzian matrices with the dual of the cut cone, a finitely generated integral polyhedral cone extensively studied in metric geometry and graph theory. We examine in detail the pentagonal ratio, which first appears for Lorentzian matrices of size at least five. For Lorentzian matrices of size three, we determine the optimal bounding constants across the entire cone of bounded ratios, obtaining an explicit entropy-like formula. We conjecture that any normalized bounded ratio is, in fact, bounded by 2.

We provide a short introduction to ``Lorentzian metric spaces" i.e., spacetimes defined solely in terms of the two-point Lorentzian distance. As noted in previous work, this structure is essentially unique if minimal conditions are imposed, such as the continuity of the Lorentzian distance and the relative compactness of chronological diamonds. The latter condition is natural for interpreting these spaces as low-regularity versions of globally hyperbolic spacetimes. Confirming this interpretation, we prove that every Lorentzian metric space admits a Cauchy time function. The proof is constructive for this general setting and it provides a novel argument that is interesting already for smooth spacetimes.

Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial sublevel sets. More precisely, let $f$ be a convex $d$-homogeneous polynomial of even degree $d \ge 2$ which is positive except at the origin. We show that the intrinsic and dual volumes of the sublevel set $[f \le 1]$ admit Laplace-type integral formulas obtained by averaging the infimal projection and restriction of $f$ over the Grassmannian. This explicit representation yields three main consequences: (1) L\"owner--John-type existence and uniqueness results extending beyond the classical volume case; (2) a block decomposition principle describing factorization of intrinsic volumes under direct-sum splitting; (3) a coordinate-free formulation of Lipschitz-type lattice discrepancy bounds. These formulas enable analytic treatment of a broad class of geometric quantities, providing direct access to variational and arithmetic applications as well as new structural insights.

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. B\"or\"oczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$.

In this note we construct a geodesic line in the Gromov-Hausdorff class for which the real line with a natural metric is an interior point.

In this note we show that the Lipschitz distance between the classes of metric spaces at finite Gromov-Hausdorff distances from the one-point metric space and the real line with the natural metric, respectively, is positive.