No Arabic abstract
Given a convex disk $K$ and a positive integer $k$, let $vartheta_T^k(K)$ and $vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $vartheta_L^k(T)=frac{2k+1}{2}$. In this paper, we will show that $vartheta_T^k(T)=vartheta_L^k(T)$.
We analyze loci of triangles centers over variants of two-well known triangle porisms: the bicentric family and the confocal family. Specifically, we evoke a more general version of Poncelets closure theorem whereby individual sides can be made tangent to separate caustics. We show that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.
We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Nasz{o}dis equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).
A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with $tinOmega(n^2)$ whose smallest such set has $1.5t-o(n^2)$ arcs.
Given a plane triangle $Delta$, one can construct a new triangle $Delta$ whose vertices are intersections of two cevian triples of $Delta$. We extend the family of operators $DeltamapstoDelta$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.