8 citations · 29 across the 18 of their papers we have counts for
22 papers
Polytopes with large transversal ratio
Michael Gene Dobbins, Seunghun Lee
The transversal ratio of a polytope is the minimum proportion of vertices of required to intersect each facet of . The weak chromatic number of is the minimum number…
Homotopy equivalence of Grassmannians and MacPhersonians in rank 3
Michael Gene Dobbins
We confirm a long standing conjecture in the case of rank 3 that MacPhersonians are homotopy equivalent to Grassmannians.
Colorful Intersections and Tverberg Partitions
Michael Gene Dobbins, Andreas F. Holmsen, Dohyeon Lee
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 fo…
Inscribable order types
Michael Gene Dobbins, Seunghun Lee
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 typ…
Transversals and colorings of simplicial spheres
Joseph Briggs, Michael Gene Dobbins, Seunghun Lee
Motivated from the surrounding property of a point set in introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simp…
A strong equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group
Michael Gene Dobbins
This is the third paper in a series on oriented matroids and Grassmannians. We construct a -equivariant strong deformation retraction from the hom…
Continuous dependence of curvature flow on initial conditions
Michael Gene Dobbins
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 flo…
The Inverse Kakeya Problem
Sergio Cabello, Otfried Cheong, Michael Gene Dobbins
We prove that the largest convex shape that can be placed inside a given convex shape in any desired orientation is the largest inscribed ball of . Th…
A Universality Theorem for Nested Polytopes
Michael G. Dobbins, Andreas Holmsen, Tillmann Miltzow
In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops and a number , the Nest…
Smoothed Analysis of the Art Gallery Problem
Michael Gene Dobbins, Andreas Holmsen, Tillmann Miltzow
In the Art Gallery Problem we are given a polygon on vertices and a number . We want to find a guard set of size , such that each point in is s…
Barycenters of points in polytope skeleta
Michael Gene Dobbins, Florian Frick
The first author showed that for a given point in an -polytope there are points in the -faces of , whose barycenter is . We show that we can increase the d…
Grassmannians and Pseudosphere Arrangements
Michael Gene Dobbins
We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements. These are defined as a…
Completeness for the Complexity Class and Area-Universality
Michael G. Dobbins, Linda Kleist, Tillmann Miltzow +1
Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class plays a crucial role in the study of geometric problems.…
Shadows of a Closed Curve
Michael Gene Dobbins, Heuna Kim, Luis Montejano +1
A shadow of a geometric object in a given direction is the orthogonal projection of on the hyperplane orthogonal to . We show that any topological embedding of a cir…
Weight Balancing on Boundaries
Luis Barba, Otfried Cheong, Michael Gene Dobbins +8
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, wh…
The Shadows of a Cycle Cannot All Be Paths
Prosenjit Bose, Jean-Lou De Carufel, Michael G. Dobbins +2
A "shadow" of a subset of Euclidean space is an orthogonal projection of into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three…
The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions
Boris Aronov, Otfried Cheong, Michael Gene Dobbins +1
We show that the union of translates of a convex body in can have holes in the worst case, where a hole in a set is a connected component of $\mathb…
Realization spaces of arrangements of convex bodies
Michael Gene Dobbins, Andreas Holmsen, Alfredo Hubard
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 mai…
A point in a -Polytope is the barycenter of points in its -faces
Michael Gene Dobbins
Using equivariant topology, we prove that it is always possible to find points in the -dimensional faces of a -dimensional convex polytope so that their center of ma…
Regular systems of paths and families of convex sets in convex position
Michael G. Dobbins, Andreas F. Holmsen, Alfredo Hubard
In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of eac…
The Erdős-Szekeres problem for non-crossing convex sets
Michael G Dobbins, Andreas F Holmsen, Alfredo Hubard
We show an equivalence between a conjecture of Bisztriczky and Fejes T{ó}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in t…
Realizability of Polytopes as a Low Rank Matrix Completion Problem
Michael Gene Dobbins
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 ma…