  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 $1 \leq i \leq n - 1$, there exists a distance which occurs exactly
$i$ times. Constructions of sizes $n \leq 8$ 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
$(\mathbb{R}^2, \| \cdot \|)$ by studying strong crescent configurations in $\|
\cdot \|$. In an arbitrary norm $\|\cdot \|$, 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 $\|\cdot
\|$. A line-like configuration in $\|\cdot \|$ 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 $\|\cdot \|$ circle. We prove that for $n \geq 7$, 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.
