Crescent configurations in normed spaces

We study the problem of crescent configurations, posed by Erd\H{o}s in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1\le i\le n-1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n\le 8$ have been provided by Liu, Pal'{a}sti, and Pomerance. Erd\H{o}s conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \geq N$. We extend the problem of crescent configurations to general normed spaces $({\mathbb{R}}^2,\mathrm{\parallel }\cdot \mathrm{\parallel })$ by studying strong crescent configurations in $| \cdot |$\mathrm{.}Inanarbitrarynorm$|\cdot |$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \leq 6$,$n \leq 8$,and$n \leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in $|\cdot |$\mathrm{.}Aline-likeconfigurationin$|\cdot |$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common $\mathrm{\parallel }\cdot \mathrm{\parallel }$ circle. We prove that for $n\ge 7$, every line-like crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.