arXiv subjects

Florian Frick

Publications and source records attributed to Florian Frick.

At least 19 recordsLinked to original sources

arXiv2026

Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel-Lindenstrauss-Milman theorem

We prove that every simplicial triangulation of real projective $d$-space has $\exp(\Omega(\sqrt d))$ vertices. Together with known constructions, this determines the minimum vertex number as $\mu_d=\exp(d^{1/2+o(1)})$. The result follows from a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: a finite strongly regular CW complex with a free cellular involution, $v$ vertices, and $f$ maximal cells has $\mathbb{Z}/2$-index at most $O(\log v\log f)$. We bound the dimensions of Morse cells by a trace estimate for a constrained Hessian, obtaining a Morse-theoretic proof of the classical inequality for centrally symmetric polytopes. As further applications of this inequality, we give an $\exp(\Omega(\sqrt t))$ lower bound for the order of a triangle-free topologically $t$-chromatic graph and bound the index of sign complexes by $O(d\log^2 N)$ for total matrices and $O(d\log^3 N)$ for partial matrices, where $N\geq2$ is the number of columns and $d\geq1$ is the VC dimension.

math.CO↗

arXiv2026

Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces

For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov--Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov--Hausdorff distance.

math.MG↗

arXiv2026

A $\mathbb{Z}_2$-Topological Framework for Sign-rank Lower Bounds

We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms. For every (partial) sign matrix $A$, we associate a free $\mathbb{Z}_2$-simplicial complex $S(A)$ and show that sign-rank of $A$ is characterized by the linear analog of $\mathbb{Z}_2$-index of $S(A)$. As a consequence, the classical $\mathbb{Z}_2$-index of $S(A)$ lower bounds the sign-rank of $A$, which reduces sign-rank lower bounds to topological obstructions. This reduction allows us to use various tools from $\mathbb{Z}_2$-equivariant topology, particularly in regimes where classical lower-bound techniques break down. As the main application, we consider the Gap Hamming Distance function $\mathrm{GHD}_k^n$ (defined for $k < n/2$), which distinguishes pairs of strings in $\{0,1\}^n$ with Hamming distance at most $k$ from pairs with distance at least $n-k$. We prove an essentially tight lower bound and show that for any $k$, \[ \text{sign-rank}(\mathrm{GHD}_k^n) = (1-o_k(1)) 2k. \] where the $o_k(1)$ term is $O\left(\sqrt{\frac{\log k}{k}}\right)$. This improves on the previous lower bound of Hatami, Hosseini, and Meng (STOC 2023) who proved that sign-rank of $\mathrm{GHD}_k^n$ is at least $\Omega(k/\log(n/k))$. A key technical ingredient is a new analysis of the $\mathbb{Z}_2$-coindex (which lower bounds $\mathbb{Z}_2$-index) of the Vietoris-Rips complex of the hypercube in the sparse regime which yields an essentially tight lower bound. Previously, no results were known in the sparse regime.

math.CO↗

arXiv2025

An optimal Brouwer's fixed point theorem for discontinuous functions

Brouwer's fixed point theorem states that any continuous function from a closed $n$-dimensional ball to itself has a fixed point. In 1961, Klee showed that if such a function has discontinuities that are bounded, then it has a point that is close to being fixed. We improve upon Klee's results in any finite-dimensional Euclidean space, and prove that our bounds are the best possible.

math.MG↗

arXiv2025

Quantifying discontinuity

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

math.MG↗

arXiv2025

Covering and labeling generalizations of the Borsuk-Ulam theorem

We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives complexity bounds in a suitable sense, whereas our extension additionally provides insight into the structure of objects satisfying the complexity bound. This structure is governed by order types of finite point sets in Euclidean space and more generally by the intersection combinatorics of faces under continuous maps from the simplex. We develop some of those applications for sphere coverings, Kneser-type colorings, Hall-type results for hypergraphs, and hyperplane mass partitions, among other consequences. We provide a new proof of the topological Hall theorem and extend it into a result that simultaneously generalizes hypergraph Hall theorems and topological lower bounds for chromatic numbers.

math.CO↗

arXiv2025

Geodesic complexity of the octahedron, and an algorithm for cut loci on convex polyhedra

The geodesic complexity of a length space $X$ quantifies the required number of case distinctions to continuously choose a shortest path connecting any given start and end point. We prove a local lower bound for the geodesic complexity of $X$ obtained by embedding simplices into $X\times X$. We additionally create and prove correctness of an algorithm to find cut loci on surfaces of convex polyhedra, as the structure of a space's cut loci is related to its geodesic complexity. We use these techniques to prove the geodesic complexity of the octahedron is four. Our method is inspired by earlier work of Recio-Mitter and Davis, and thus recovers their results on the geodesic complexity of the $n$-torus and the tetrahedron, respectively.

math.MG↗

arXiv2025

Roots of real-valued zero mean maps: Compositions of linear functionals and equivariant maps

We develop a novel topological framework that yields results constraining the distribution of zeros of certain zero mean real-valued maps, namely those obtained from composing a fixed equivariant map with linear functionals. We use this framework to establish upper bounds for the topology of set systems in the domain where (multivariate) trigonometric polynomials do not change their sign, generalizing and, in certain regimes, strengthening results in the literature. Our results more generally contain restrictions on the distribution of zeros of Chebyshev spaces as special cases. Lastly, we apply this framework to derive existence results for efficient cubature rules for compositions of affine functionals and equivariant maps.

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arXiv2024

Youden's Demon is Sylvester's Problem

If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $\frac 6 \pi \arcsin\left(\frac{1}{3}\right)\approx .649$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line ("Youden's demon problem") is the same as a natural generalization of Sylvester's Four Point problem to Gaussian points in $\mathbb{R}^d$. Our main tool is the observation that the Gale dual of independent samples in $\mathbb{R}^d$ itself can be taken to be a set of independent points (conditioned on barycenter at the origin) when the distribution of the points is Gaussian.

math.PR↗

arXiv2023

Topological methods in zero-sum Ramsey theory

A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in $\mathbb{Z}/n$ contains a zero-sum subsequence of length $n$. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any $\mathbb{Z}/n$-coloring of an $n$-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson's generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

math.CO↗

arXiv2023

Hausdorff vs Gromov-Hausdorff distances

Let $M$ be a closed Riemannian manifold and let $X\subseteq M$. If the sample $X$ is sufficiently dense relative to the curvature of $M$, then the Gromov-Hausdorff distance between $X$ and $M$ is bounded from below by half their Hausdorff distance, namely $d_{GH}(X,M) \ge \frac{1}{2} d_H(X,M)$. The constant $\frac{1}{2}$ can be improved depending on the dimension and curvature of the manifold $M$, and obtains the optimal value $1$ in the case of the unit circle, meaning that if $X\subseteq S^1$ satisfies $d_{GH}(X,S^1)<\tfrac{\pi}{6}$, then $d_{GH}(X,S^1)=d_H(X,S^1)$. We also provide versions lower bounding the Gromov-Hausdorff distance $d_{GH}(X,Y)$ between two subsets $X,Y\subseteq M$. Our proofs convert discontinuous functions between metric spaces into simplicial maps between \v{C}ech or Vietoris-Rips complexes. We then produce topological obstructions to the existence of certain maps using the nerve lemma and the fundamental class of the manifold, thus lower bounding the Gromov-Hausdorff distance.

math.MG↗

arXiv2023

Colorful Borsuk--Ulam theorems and applications

We prove a colorful generalization of the Borsuk--Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case. We prove a colorful generalization of Fan's antipodal sphere covering theorem, we derive a short proof of Gale's colorful KKM theorem, and we prove a colorful generalization of Brouwer's fixed point theorem. Our results also provide an alternative between Radon-type intersection results and KKM-type covering results. Finally, we prove colorful Borsuk--Ulam theorems for higher symmetry.

math.CO↗

arXiv2022

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{\'e}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.

math.MG↗

arXiv2022

Transversal generalizations of hyperplane equipartitions

The classical Ham Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asserts that any $d$ families of pairwise intersecting compact, convex sets in $\mathbb{R}^d$ admit a common hyperplane transversal. We extend Dolnikov's theorem by showing that families of compact convex sets satisfying more general non-disjointness conditions admit common transversals by multiple hyperplanes. In particular, these generalize all known optimal results to the long-standing Gr\"unbaum--Hadwiger--Ramos measure equipartition problem in the case of two hyperplanes. Our proof proceeds by establishing topological Radon-type intersection theorems and then applying Gale duality in the linear setting. For a single hyperplane, this gives a new proof of Dolnikov's original result via Sarkaria's non-embedding criterion for simplicial complexes.

math.CO↗

arXiv2022

Unit sphere fibrations in Euclidean space

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.

math.GT↗

arXiv2022

Random complexes with free involution

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural result for pairs of non-adjacent cliques in Erd\H{o}s--R\'{e}nyi random graphs.

math.CO↗

arXiv2022

Vietoris thickenings and complexes have isomorphic homotopy groups

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris complex contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, equipped with an optimal transport metric. We show that the Vietoris metric thickening and the Vietoris complex have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover $\mathcal{U}$ appropriately, we get isomorphisms between the homotopy groups of Vietoris--Rips metric thickenings and simplicial complexes, where both spaces are defined using the convention ``diameter $< r$'' (instead of $\le r$). Similarly, we get isomorphisms between the homotopy groups of \v{C}ech metric thickenings and simplicial complexes, where both spaces are defined using open balls (instead of closed balls).

math.MG↗

arXiv2022

Colorful Words and d-Tverberg Complexes

We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb R^d$. This strengthens the concept of $d$-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed $d$ a graph that is not weakly $d'$-Tverberg for any $d'\le d$, answering a question of De Loera, Hogan, Oliveros, and Yang.

math.CO↗