: Mathematics: Minkowski functional

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Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If ${\\displaystyle K}$ is a subset of a real or complex vector space ${\\displaystyle X,}$ then the Minkowski functional or gauge of ${\\displaystyle K}$ is defined to be the function ${\\displaystyle p_{K}:X\ o [0,\\infty ],}$ valued in the extended real numbers, defined by
${\\displaystyle p_{K}(x):=\\inf\\{r\\in \\mathbb {R} :r>0{\ ext{ and }}x\\in rK\\}\\quad {\ ext{ for every }}x\\in X,}$
where the infimum of the empty set is defined to be positive infinity ${\\displaystyle \\,\\infty \\,}$ (which is not a real number so that ${\\displaystyle p_{K}(x)}$ would then not be real-valued).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Minkowski_functional)

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