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. The set K {\textstyle K} is often assumed to have properties, such as being an absorbing disk in X {\textstyle X} , that guarantee that p K {\textstyle p_{K}} will be a seminorm on X . {\textstyle X.} In fact, every seminorm p {\textstyle p} on X {\textstyle X} is equal to the Minkowski functional (that is, p = p K {\textstyle p=p_{K}} ) of any subset K {\textstyle K} of X {\textstyle X} satisfying { x ∈ X : p ( x ) < 1 } ⊆ K ⊆ { x ∈ X : p ( x ) ≤ 1 } {\displaystyle \{x\in X:p(x)<1\}\subseteq K\subseteq \{x\in X:p(x)\leq 1\}} (where all three of these sets are necessarily absorbing in X {\textstyle X} and the first and last are also disks). Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can...