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Register a new userA family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by nine points
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Daniel McGinnis
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We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gyarfas, Kleitman, and Toth in 2001.
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Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed. We achieve this by constructing a convex realization for an arbitrary code with $k$ nonempty codewords in $mathbb{R}^{k-1}$. This result justifies the usual restriction of the definition of convex neural codes to include only those that can be realized by receptive fields that are all either open convex or closed convex. We also show that the dimension of our construction cannot in general be lowered.
Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation.
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PolyMath REU Convex Geometries Collaboration: Kira Adaricheva
, Madinan Bolat
, Gent Gjonbalaj
2020
A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we made representations of 621. Of the 51 remaining geometries on a 5-element set, one was already shown not to be representable due to the Weak Carousel property, as articulated by Adaricheva and Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more of these convex geometries cannot be represented by circles on the plane, due to what we term the Triangle Property.
We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result, under weakened conditions on the sets. A triple of sets $A,B,C$ in the plane is said to be a {em tight} if $textrm{conv}(Acup B)cap textrm{conv}(Acup C)cap textrm{conv}(Bcap C) eq emptyset.$ This notion was first introduced by Holmsen, where he showed that if $mathcal{F}$ is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least $frac{1}{8}|mathcal{F}|$ members of $mathcal{F}$. Here we prove that if $mathcal{F}_1,dots,mathcal{F}_6$ are families of compact connected sets in the plane such that every three sets, chosen from three distinct families $mathcal{F}_i$, form a tight triple, then there exists $1le jle 6$ and three lines intersecting every member of $mathcal{F}_j$. In particular, this improves $frac{1}{8}$ to $frac{1}{3}$ in Holmsens result.
The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight over all points that they obtain. We restrict the setting to the case of binary weights. We show a construction of an arbitrarily large odd-sized point set that allows Bob to obtain almost 3/4 of the total weight. This construction answers a question asked by Matsumoto, Nakamigawa, and Sakuma in [Graphs and Combinatorics, 36/1 (2020)]. We also present an arbitrarily large even-sized point set where Bob can obtain the entirety of the total weight. Finally, we discuss conjectures about optimum moves in the convex grabbing game for both players in general.
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