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 ${\displaystyle p_{K}:X\to [0,\infty ],}$ valued in the extended real numbers, defined by $${\displaystyle p_{K}(x):=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}\quad {\text{ for every }}x\in X,}$$ 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).

The set ${\displaystyle K}$ is often assumed/picked to have properties, such as being an absorbing disk in ${\displaystyle X,}$ that guarantee that ${\displaystyle p_{K}}$ will be a real-valued seminorm on ${\displaystyle X.}$ In fact, every seminorm ${\displaystyle p}$ on ${\displaystyle X}$ is equal to the Minkowski functional (that is, ${\displaystyle p=p_{K}}$) of any subset ${\displaystyle K}$ of ${\displaystyle X}$ satisfying ${\displaystyle \{x\in X:p(x)<1\}\subseteq K\subseteq \{x\in X:p(x)\leq 1\}}$ (where all three of these sets are necessarily absorbing in ${\displaystyle X}$ 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 be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of ${\displaystyle X}$ into certain algebraic properties of a function on ${\displaystyle X.}$

The Minkowski function is always non-negative (meaning ${\displaystyle p_{K}\geq 0}$). This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values. However, ${\displaystyle p_{K}}$ might not be real-valued since for any given ${\displaystyle x\in X,}$ the value ${\displaystyle p_{K}(x)}$ is a real number if and only if ${\displaystyle \{r>0:x\in rK\}}$ is not empty. Consequently, ${\displaystyle K}$ is usually assumed to have properties (such as being absorbing in ${\displaystyle X,}$ for instance) that will guarantee that ${\displaystyle p_{K}}$ is real-valued.