Point sets containing their triangle centers

Let S be a set of at least five points in the plane, not all on a line. Suppose that for any three points $${a,b,c\in S}$$ the nine-point center of triangle abc also belongs to S. We show that S must be dense in the plane. We also consider several problems about partitioning the plane into two sets containing their triangle center

Diameter Bounds for Planar Graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight

Large simplices determined by finite point sets

Given a set P of n points in $$\mathbb R ^{d}$$ , let $${d}_{1}>d_{2}>\cdots$$ denote all distinct inter-point distances generated by point pairs in $$P$$ . It was shown by Schur, Martini, Perles, and Kupitz that there is at most one d-dimensional regular simplex of edge length $${d}_{1}$$ whose every vertex belongs to P. We extend this result by showing that for any k the number of d-dimensional regular simplices of edge length $${d}_{k}$$ generated by the points of P is bounded from above by a constant that depends only on d and

On Polygons Excluding Point Sets

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points $R$. We show that there is a minimal number $K=K(l)$, which is polynomial in $l$, such that one can always find a blue polygonization excluding all red points, whenever $k\geq K$. Some other related problems are also considered.Comment: 14 pages, 15 figure

Upper bounds for the perimeter of plane convex bodies

We show that the maximum total perimeter ofk plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body

On Polygons Excluding Point Sets

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that $${B\cup R}$$ is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado etal. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K=K(l), which is bounded from above by a polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k≥ K. Some other related problems are also considere

Supplementary data for article: Aleksić, I.; Šegan, S.; Andrić, F.; Zlatović, M.; Moric, I.; Opsenica, D. M.; Senerovic, L. Long-Chain 4-Aminoquinolines as Quorum Sensing Inhibitors in Serratia Marcescens and Pseudomonas Aeruginosa. ACS Chemical Biology 2017, 12 (5), 1425–1434. https://doi.org/10.1021/acschembio.6b01149

Supporting information for: [https://doi.org/10.1021/acschembio.6b01149]Related to published version: [http://cherry.chem.bg.ac.rs/handle/123456789/2461]Related to accepted version: [http://cherry.chem.bg.ac.rs/handle/123456789/3089

Supplementary data for article: Aleksić, I.; Šegan, S.; Andrić, F.; Zlatović, M.; Moric, I.; Opsenica, D. M.; Senerovic, L. Long-Chain 4-Aminoquinolines as Quorum Sensing Inhibitors in Serratia Marcescens and Pseudomonas Aeruginosa. ACS Chemical Biology 2017, 12 (5), 1425–1434. https://doi.org/10.1021/acschembio.6b01149

Supporting information for: [https://doi.org/10.1021/acschembio.6b01149]Related to published version: [http://cherry.chem.bg.ac.rs/handle/123456789/2461]Related to accepted version: [http://cherry.chem.bg.ac.rs/handle/123456789/3089