Minkowski functional

is a subset of a real or complex vector spacevalued in the extended real numbers, defined bywhere the infimum of the empty set is defined to be positive infinityThus 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).In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset ofbe a subset of a real or complex vector spacevalued in the extended real numbers, defined bybe a vector space without topology with underlying scalar fieldThis is characteristic of Minkowski functionals defined via "nice" sets.There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets.What is meant precisely by "nice" is discussed in the section below.Notice that, in contrast to a stronger requirement for a norm,is convex and the origin belongs to the algebraic interior of[3] Arguably the most common requirements placed on a setDue to how common these assumptions are, the properties of a Minkowski functionalis a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and letThis section will investigate the most general case of the gauge of any subsetis a subset of a real or complex vector spaceis straightforward and can be found in the article on absorbing sets.This proves that Minkowski functionals are strictly positive homogeneous.The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity ofthat have a certain purely algebraic property that is commonly encountered.Only (1) implies (3) will be proven because afterwards, the rest of the theorem follows immediately from the basic properties of Minkowski functionals described earlier; properties that will henceforth be used without comment.For instance, it can be used to describe how every real homogeneous functionbe a subset of a real or complex vector spaceis necessarily convex, balanced, absorbing, and satisfiesis a convex, balanced, and absorbing subset of a real or complex vector spaceon an arbitrary topological vector spaceis a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers.where this convex open neighborhood of the origin satisfies