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 K {\textstyle K} is a subset of a real or complex vector space X , {\textstyle X,} then the Minkowski functional or gauge of K {\textstyle K} is defined to be the function p K : X → [ 0 , ∞ ] , {\textstyle p_{K}:X\to [0,\infty ],} valued in the extended real numbers, defined by p K ( x ) = inf { r ∈ R : r > 0 and x ∈ r K } , x ∈ X , {\displaystyle p_{K}(x)=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\},\quad x\in X,} where the infimum of the empty set is defined to be positive infinity.