## The Paris metric with an open ball

### Task number: 3166

The Paris metric $$\rho_P$$ (known also as the mailman metric) on $$\mathbb R^2$$ is defined as follows:

Let $$o$$ denote the origin of coordinates, i.e. the point $$(0, 0)$$.

If $$x, y$$ lie on the same spoke (semi line) originating at $$o$$, then $\rho_P(x, y) = \| x - y\|,$ where $$\|x - y\|$$ is the Euclidean distance between $$x$$ a $$y$$.

If $$x, y$$ do not lie on the same spoke originating at $$o$$, then $\rho_P(x, y) = \| x - o\| + \|o - y\|.$

• #### Variant 1

Verify that $$(\mathbb R^2, \rho_P)$$ is a metric space.

• #### Variant 2

Draw the open ball with center $$(1{,}1)$$ and radius $$2$$ in $$(X, \rho_P)$$.