## The New York metric with an open ball

### Task number: 3167

The New York metric \(\rho_N\) (known also as the \(L_1\) norm) on \(\mathbb R^2\) is defined as \[ \rho((x_1, x_2),(y_1, y_2)) = |x_1 - y_1| + |x_2 - y_2|, \] kde \((x_1, x_2), (y_1, y_2) \in \mathbb R^2\).

#### Variant 1

Show that \((\mathbb R^2, \rho_N)\) is a metric space.

#### Variant 2

Draw the open ball with center \((1, 1)\) and radius \(2\) v \((\mathbb R^2, \rho_N)\).