Xuemei He, Liping Yuan, Tudor Zamfirescu: $rq$-Convexity of lattice graphs, 25-34

Abstract:

Let $\{x,y,w,z\}\subset {\Bbb R}^{d}$. If $\mathrm{conv}\{x,y,w,z\}$ is a non-degenerate rectangle, then we call the set $\{x,y,w,z\}$ a rectangular quadruple. Let $M \subset {\Bbb R}^{d}$ with $\mathrm{card}M\geq 4$. If, for any $x,y\in M$, there exists a rectangular quadruple $\{x,y,w,z\}\subset M$, we say that $M$ is $rq$-convex and the pair $x,y$ have the $rq$-property in $M$. In this paper, we consider $rq$-convexity of lattice graphs which are in the planar square and triangular lattices and the cubic lattice in $3$-space.

Key Words: Rectangular quadruple, $rq$-convexity, lattice graphs, cubic lattice.

2020 Mathematics Subject Classification: Primary 52A01; Secondary 52C20.

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