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 \((\mathbb R^2, \rho_N)\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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