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 \((X, \rho_P)\).

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