The thinnest holding-lattice of a set (Q578593)

We study the following question proposed by F. Fejes Tóth in personal communication. In the Euclidean plane a lattice \(\Gamma\) is called a holding-lattice of a planar set S if any set congruent to S contains at least one lattice point of \(\Gamma\). The density of \(\Gamma\) equals 1/(2\(\Delta)\), where \(\Delta\) denotes the area of a fundamental triangle of \(\Gamma\). A holding-lattice of S of least possible density is said to be the thinnest holding-lattice of S. The problem which is answered in many cases in this paper is now the following: Find the sets whose thinnest holding-lattice is a regular triangle-lattice.