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Minkowski functional
Function made from a set / From Wikipedia, the free encyclopedia
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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.
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If is a subset of a real or complex vector space
then the Minkowski functional or gauge 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 not a real number so that
would then not 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 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 into certain algebraic 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.