CyberSec Research

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski chirality'' measures to Banach--Mazur distances and to each other, and prove their continuity with respect to the Hausdorff distance. In the planar case, we determine the reflection axes at which the Minkowski chirality of triangles and parallelograms is attained, and show that $\sqrt{2}$ is a tight upper bound on the chirality in both cases.

Where are the intersection points of diagonals of a regular $n$-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

In this paper, we provided $L_p$ curvature measures of affine dual quermassintegrals for $p\in\mathbb{R}$, and solved the existence part of the $L_p$ Minkowski problems for the non-symmetric measures when $p>1$ and symmetric measures when $p\geq0$. When $p=0$, this is the affine dual Minkowski problems, which is introduced and solved by Cai-Leng-Wu-Xi in [7].

Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric graphs as abstract tropical curves remains unexplored. In this paper, we present the first computational and machine learning-driven study of metric graphs from the perspective of tropical algebraic geometry. Specifically, we study the tropical Abel--Jacobi transform, a vectorization of points on a metric graph via the tropical Abel--Jacobi map into its associated flat torus, the tropical Jacobian. We develop algorithms to compute this transform and investigate how the resulting embeddings depend on different combinatorial models of the same metric graph. Once embedded, we compute pairwise distances between points in the tropical Jacobian under two natural metrics: the tropical polarization distance and the Foster--Zhang distance. Computing these distances are generally NP-hard as they turn out to be linked to classical lattice problems in computational complexity, however, we identify a class of metric graphs where fast and explicit computations are feasible. For the general case, we propose practical algorithms for both exact and approximate distance matrix computations using lattice basis reduction and mixed-integer programming solvers. Our work lays the groundwork for future applications of tropical geometry and the tropical Abel--Jacobi transform in machine learning and data analysis.

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erd\H{o}s/Falconer type problems.

For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for the maximum cardinality $\alpha(n, t)$ of such a set based on an extension of the Lov\'asz theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of $n$ and $t$ it is better. We also refine existing spectral methods to show that $\alpha(n, t) \leq 2(n+1)$ for all $n$ and $t \leq 0$, with equality only at $t = -1/n$. This allows us to show the uniqueness of the optimal construction at $t = -1/n$ for $n \leq 5$ and to enumerate all possible constructions for $n \leq 3$ and $t \leq 0$.

In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, for $p \in \mathbb{R}$ such that $p>m$, in Alexandrov spaces of dimension $n$ with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension $n$ whose metric tensor is at least of class $\mathcal{C}^2$, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on $n$, $m$, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the $\mathcal{C}^{1,\alpha}$ topology, for every $\alpha \in ]0,1[$ (in fact, it is enough with the $W^{2,q}$ topology for some suitable $q$ large enough).

To goal of the paper is to introduce a convergence \`a la Gromov-Hausdorff for Lorentzian spaces, building on $\epsilon$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality. The final part of the paper establishes several applications: we show that Chru\'sciel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

For $p\in(1,+\infty)$, we prove that a metric measure space endowed with a $p$-energy satisfies the chain condition, the volume regular condition with respect to a doubling scaling function $\Phi$, and that both the Poincar\'e inequality and the capacity upper bound with respect to a doubling scaling function $\Psi$ hold if and only if $$\frac{1}{C}\left(\frac{R}{r}\right)^p\le\frac{\Psi(R)}{\Psi(r)}\le C\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}\text{ for any }r\le R.$$ In particular, given any pair of doubling functions $\Phi$ and $\Psi$ satisfying the above inequality, we construct a metric measure space endowed with a $p$-energy on which all the above conditions are satisfied. As a direct corollary, we prove that a metric measure space is $d_h$-Ahlfors regular and has $p$-walk dimension $\beta_p$ if and only if $$p\le\beta_p\le d_h+(p-1).$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan (arXiv:2410.15611).

The result of Guan and Ma (Invent. Math. 151 (2003)) states that if $\phi^{-1/k} : \mathbb{S}^n \to (0,\infty)$ is spherically convex, then $\phi$ arises as the $\sigma_k$ curvature (the $k$-th elementary symmetric function of the principal radii of curvature) of a strictly convex hypersurface. In this paper, we establish an analogous result in the capillary setting in the half-space for $\theta\in(0,\pi/2)$: if $\phi^{-1/k} : \mathcal{C}_{\theta} \to (0,\infty)$ is a capillary function and spherically convex, then $\phi$ is the $\sigma_k$ curvature of a strictly convex capillary hypersurface.

In this paper, we consider two limiting cases ($\alpha\rightarrow n$ and $\alpha\rightarrow 0 $) of the recent affine HLS inequalities by Haddad and Ludwig. As $\alpha\rightarrow n$, the affine logarithmic HLS inequality is established, which is stronger than the logarithmic HLS inequality by Carlen and Loss from 1992 and Beckner from 1993. As $\alpha\rightarrow 0$, an affine version of Beckner's logarithmic Sobolev inequality is established, which is also a limiting case of the affine fractional $L^2$ Sobolev inequalities. The affine logarithmic Sobolev inequality is stronger than the original version by Beckner from 1995.

We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear programming algorithms to enumerate all feasible interlacing polynomials of a given polynomial that comes from a certain family of real-rooted integer polynomials. These algorithms are further specialised for the study of real equiangular lines, incorporating additional number-theoretic constraints to restrict the enumeration. Our improvements significantly enhance the efficiency of the methods presented in previous work by the authors.

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus in 1999 from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively.

The structure bilateral trading costs is one of the key features of international trade. Drawing upon the freeness-of-trade matrix, which allows the modeling of N-state trade costs, we develop a ``geometry of inconvenience'' to better understand how they impact equilbrium outcomes. The freeness-of-trade matrix was introduced in a model by Mossay and Tabuchi, where they essentially proved that if a freeness-of-trade matrix is positive definite, then the corresponding model admits a unique equilibrium. Drawing upon the spectral theory of metrics, we prove the model admits nonunique, perverse, equilibria. We use this result to provide a family of policy relevant bipartite examples, with substantive applications to economic sanctions. More generally, we show how the network structure of the freeness of trade is central to understanding the impacts of policy interventions.

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $\pi_1(M)$. If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$ and additionally the Ricci curvature is asymptotically non-negative, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales. If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $n-2$-Ricci curvature is asymptotically non-negative, then we prove that the dimension drop at large scales improves to $n-2$. From the above results, we deduce topological restrictions, such as upper bounds on the first Betti number.

We generalize classical kinematic formulas for convex bodies in a real vector space $V$ to the setting of non-compact Lie groups admitting a Cartan decomposition. Specifically, let $G$ be a closed linear group with Cartan decomposition $G \cong K \times \exp(\mathfrak{p}_0)$, where $K$ is a maximal compact subgroup acting transitively on the unit sphere. For $K$-invariant continuous valuations on convex bodies, we establish an integral geometric-type formula for $\overline{G} = G \ltimes V$. Key to our approach is the introduction of a Gaussian measure on $\mathfrak{p}_0$, which ensures convergence of the non-compact part of the integral. In the special case $K = O(n)$, we recover a Hadwiger-type formula involving intrinsic volumes, with explicit constants $c_j$ computed via a Weyl integration formula.

We prove that the existence of an affine map on a Hadamard space which is not a dilation, characterizes Riemannian symmetric spaces of higher rank, Euclidean buildings of higher rank and non-trivial metric products.

We prove the timelike Brunn-Minkowski inequality $\mathsf{TBM}(K,N)$ implies a timelike lower bound on the Bakry-\'Emery-Ricci curvature on weighted globally hyperbolic spacetimes. This result, together with the well-known equivalence between timelike Bakry-\'Emery-Ricci lower bounds and the $\mathsf{TCD}(K,N)$ condition, and the fact that $\mathsf{TCD}(K,N)$ spaces support the timelike Brunn-Minkowski inequality, draws an equivalence between $\mathsf{TBM}(K,N)$ and $\mathsf{TCD}(K,N)$ in the smooth setting.