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  "body": "\n## Example\n\nLet $n \\ge 1$ be an integer and let $p \\ge 1$ be a real number.\nFor $x = (x_1, \\ldots, x_n)$ and $y = (y_1, \\ldots, y_n)$ in $\\mathbb{R}^n$, let\n$$d_p(x, y) = \\left( \\sum_{i=1}^{n} |x_i - y_i|^p \\right)^{1/p}.$$\nThen $d_p$ is a [[metric|063W5JMY85]] on $\\mathbb{R}^n$.\n\n## Remarks\n\nThe triangle inequality for $d_p$ is the Minkowski inequality.\nFor $p = 1$ it is the [[taxicab metric|07CJNE6ZAQ]], and for $p = 2$ it is the [[Euclidean metric|0E1MMCAY5B]].\nFor $0 \u003c p \u003c 1$, the same formula does not give a metric when $n \\ge 2$: the triangle inequality fails.\nThe metric $d_p$ comes from the [[p-norm|0KNJWEVSMG]]: $d_p(x, y) = \\|x - y\\|_p$.\n\n## Explanations\n\n### L1 human\n\nThe p-metrics form a family of ways to combine the differences of the coordinates into one distance.\nA larger $p$ gives more weight to the largest difference, and for very large $p$ the distance is close to the [[sup metric|0VTAXQEWQD]].\n",
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