Shape of a unit Ball and metric Playing around with metrics defined in $\Bbb{R}^2$, such as $d_p$, where $p\ge1$, or even such as $σ=\sqrt{d_p}, τ=\frac{d_p}{1+d_p},π=\min \{1,d_p\}$, I realised that all produce sets in $\Bbb{R}^2$ ("unit balls") that are convex sets: .
So I wonder if there is a metric in $\Bbb{R}^2$, such as its respective unit ball is not convex, or not simply connected.
Thanks a lot.
 A: $\newcommand{\Vec}[1]{\mathbf{#1}}$Suppose a metric $d$ is induced by a norm. The $d$-unit ball is convex by homogeneity and the triangle inequality: If the unit ball is not convex, then there exist points $\Vec{x}$ and $\Vec{y}$ a real $t$ with $0 < t < 1$ such that $|\Vec{x}| < 1$ and $|\Vec{y}| < 1$ but $|(1 - t)\Vec{x} + t\Vec{y}| \geq 1$, so either homogeneity or the triangle inequality fails.

For a "roundhouse-type" metric
$$
d(\Vec{x}, \Vec{y}) = \begin{cases}
  |\Vec{x} - \Vec{y}| & \text{if $\Vec{x}$, $\Vec{y}$ lie on a line through the origin,} \\
  |\Vec{x}| + |\Vec{y}| & \text{otherwise},
\end{cases}
$$
a ball can fail to be convex. For example, the unit ball about $\Vec{x}_{0} := (\frac{3}{4}, 0)$ is the union of the ray from $\Vec{x}_{0} + (1, 0) = (\frac{7}{4}, 0)$ to the origin and the open Euclidean disk of radius $\frac{1}{4}$ about the origin.
(The distance function gives the Euclidean distance along lines through the origin. For points on different lines, the distance is the Euclidean length of the path from $\Vec{x}$ to $\Vec{0}$ to $\Vec{y}$ along rays.)
To find a metric for which some unit ball about the origin is not simply-connected, think of the graph of a smooth function with a tall spike at a point close to the origin, such as
$$
f(x, y) = 10e^{-10,000[(x - 0.1)^{2} + y^{2}]},
$$
and define $d(\Vec{x}, \Vec{y})$ to be the infimum of the Euclidean arc lengths of smooth paths on the graph joining the point over $\Vec{x}$ to the point over $\Vec{y}$. The spike in the graph makes traveling close to the center of the spike "costly" in terms of distance, so the unit ball oozes around the spike, giving an annulus.
