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 is a subset of a real or complex vector space then the or of is defined to be the function valued in the extended real numbers, defined by where the infimum of the empty set is defined to be positive infinity (which is a real number so that would then be real-valued). The set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real-valued seminorm on In fact, every seminorm on is equal to the Minkowski functional (that is, ) of any subset of satisfying (where all three of these sets are necessarily absorbing in and the first and last are also disks). Thus every seminorm (which is a defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a 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 properties of a subset of into certain properties of a function on The Minkowski function is always non-negative (meaning ). 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, might not be real-valued since for any given the value is a real number if and only if is not empty. Consequently, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that is real-valued. Let be a subset of a real or complex vector space Define the of or the associated with or induced by as being the function valued in the extended real numbers, defined by where recall that the infimum of the empty set is (that is, ).

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