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Michael Gene Dobbins

Publications and source records attributed to Michael Gene Dobbins.

18 recordsLinked to original sources
arXiv2026

Polytopes with large transversal ratio

The transversal ratio of a polytope $P$ is the minimum proportion of vertices of $P$ required to intersect each facet of $P$. The weak chromatic number of $P$ is the minimum number of colors required to color the vertices of $P$ so that no facet is monochromatic. We will construct an infinite family of $d$-polytopes for each $d\geq 5$ whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for $d$-polytopes is unbounded for each $d\geq 5$. The previous best known lower bounds on the supremum of the transversal ratio for $d$-polytopes for $d\geq 5$ were 2/5 for odd $d$ by Novik and Zheng, and 1/2 for even $d$ by Holmsen, Pach, and Tverberg. In the case of simplicial $(d-1)$-spheres, the best known lower bounds were 1/2 for $d=5$ and $6/11$ for $d=6$ by Novik and Zheng.

math.CO↗
arXiv2024

Colorful Intersections and Tverberg Partitions

The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in $\mathbb{R}^d$, each of size $n > (\frac{d}{m}+1)(k-1)$, such that for any choice $C_i\in F_i$ we have $\bigcap_{i=1}^mC_i\neq \emptyset$. Then, one of the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$ can be partitioned into $k$ nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning $r$-dimensional transversals to families of convex sets in $\mathbb{R}^d$ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.

math.CO↗
arXiv2023

Inscribable order types

We call an order type inscribable if it is realized by a point configuration where the extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable, and that the number of such order types is $Θ(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability.

math.MG↗
arXiv2023

Transversals and colorings of simplicial spheres

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infintely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathsf{K})$ of a $d$-dimensional simplicial sphere $\mathsf{K}$ has the chromatic number $χ(\mathcal{F}(\mathsf{K})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathsf{K}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz.

math.CO↗
arXiv2021

Weight Balancing on Boundaries

Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $\lfloor d/2 \rfloor$-face and a point on a $\lceil d/2\rceil$-face.

cs.CG↗
arXiv2021

A strong equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group

This is the third paper in a series on oriented matroids and Grassmannians. We construct a $(\mathrm{O}_3\times\mathbb{Z}_2)$-equivariant strong deformation retraction from the homeomorphism group of the 2-sphere to $\mathrm{O}_3$, where the action of $\mathbb{Z}_2$ is generated by antipodal reflection acting on the right, and $\mathrm{O}_3$ acts on the left by isometry. Quotienting by the antipodal map induces a $\mathrm{SO}_3$-equivariant strong deformation retraction from the homeomorphism group of the projective plane to $\mathrm{SO}_3$. The same holds for subgroups of homeomorphisms that preserve the system of null sets. This confirms a conjecture of Mary-Elizabeth Hamstrom.

math.GT↗
arXiv2021

Continuous dependence of curvature flow on initial conditions

We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Fréchet distance in the case where the curve bisects the sphere. This even holds in the limit as time goes to infinity. This builds on Joseph Lauer's work on existence and uniqueness of solutions to the curvature flow problem on the sphere when the initial curve is not smooth.

math.DG↗
arXiv2019

The Inverse Kakeya Problem

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest volume" and when it means "largest surface area". The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

math.MG↗
arXiv2019

Grassmannians and Pseudosphere Arrangements

We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements. These are defined as a weighted variant of arrangements of pseudospheres, as in the Topological Representation Theorem for oriented matroids. We show that in rank 3, the real Stiefel manifold, Grassmannian, and oriented Grassmannian are homotopy equivalent to the analogously defined spaces of weighted pseudosphere arrangements. As a consequence, this gives a new classifying space for rank 3 vector bundles and for rank 3 oriented vector bundles where the difficulties of real algebraic geometry that arrise in the Grassmannian can be avoided. In particular, we show for all rank 3 oriented matroids, that the subspace of weighted pseudosphere arrangements realizing that oriented matroid is contractible. This is a sharp contrast with vector configurations where the space of realizations can have the homotopy type of any primary real semialgebraic set.

math.MG↗
arXiv2018

Smoothed Analysis of the Art Gallery Problem

In the Art Gallery Problem we are given a polygon $P\subset [0,L]^2$ on $n$ vertices and a number $k$. We want to find a guard set $G$ of size $k$, such that each point in $P$ is seen by a guard in $G$. Formally, a guard $g$ sees a point $p \in P$ if the line segment $pg$ is fully contained inside the polygon $P$. The history and practical findings indicate that irrational coordinates are a "very rare" phenomenon. We give a theoretical explanation. Next to worst case analysis, Smoothed Analysis gained popularity to explain the practical performance of algorithms, even if they perform badly in the worst case. The idea is to study the expected performance on small perturbations of the worst input. The performance is measured in terms of the magnitude $δ$ of the perturbation and the input size. We consider four different models of perturbation. We show that the expected number of bits to describe optimal guard positions per guard is logarithmic in the input and the magnitude of the perturbation. This shows from a theoretical perspective that rational guards with small bit-complexity are typical. Note that describing the guard position is the bottleneck to show NP-membership. The significance of our results is that algebraic methods are not needed to solve the Art Gallery Problem in typical instances. This is the first time an $\exists\mathbb{R}$-complete problem was analyzed by Smoothed Analysis.

cs.CG↗
arXiv2018

Barycenters of points in polytope skeleta

The first author showed that for a given point $p$ in an $nk$-polytope $P$ there are $n$ points in the $k$-faces of $P$, whose barycenter is $p$. We show that we can increase the dimension of $P$ by $r$, if we allow $r$ of the points to be in $(k+1)$-faces. While we can force points with a prescribed barycenter into faces of dimensions $k$ and $k+1$, we show that the gap in dimensions of these faces can never exceed one. We also investigate the weighted analogue of this question, where a convex combination with predetermined coefficients of $n$ points in $k$-faces of an $nk$-polytope is supposed to equal a given target point. While weights that are not all equal may be prescribed for certain values of $n$ and $k$, any coefficient vector that yields a point different from the barycenter cannot be prescribed for fixed $n$ and sufficiently large $k$.

math.MG↗
arXiv2017

Shadows of a Closed Curve

A shadow of a geometric object $A$ in a given direction $v$ is the orthogonal projection of $A$ on the hyperplane orthogonal to $v$. We show that any topological embedding of a circle into Euclidean $d$-space can have at most two shadows that are simple paths in linearly independent directions. The proof is topological and uses an analog of basic properties of degree of maps on a circle to relations on a circle. This extends a previous result which dealt with the case $d=3$.

math.MG↗
arXiv2015

Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindstöm. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial "stamp" polytope with a specified face that is projectively equivalent to the given polytope in all realizations. Here we construct a new stamp polytope that is closely related to Richter-Gebert's proof of universality for 4-dimensional polytopes, and we generalize several tools from that proof.

math.MG↗
arXiv2015

Realization spaces of arrangements of convex bodies

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.

math.MG↗
arXiv2015

The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions

We show that the union of $n$ translates of a convex body in $\mathbb{R}^3$ can have $Θ(n^3)$ holes in the worst case, where a hole in a set $X$ is a connected component of $\mathbb{R}^3 \setminus X$. This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.

cs.CG↗
arXiv2014

A point in a $nd$-Polytope is the barycenter of $n$ points in its $d$-faces

Using equivariant topology, we prove that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold Minkowski sum of a $nd$-polytope's $d$-skeleton is that polytope scaled by $n$. This verifies a conjecture by Takeshi Tokuyama.

math.MG↗
arXiv2012

Realizability of Polytopes as a Low Rank Matrix Completion Problem

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope.

math.CO↗