Minkowski functional

rdf:langString Minkowski functional

rdf:langString Summary

rdf:langString Theorem

rdf:langString Corollary

rdf:langString Proposition

rdf:langString true

rdf:langString Proof

rdf:langString Proof that the Gauge of an absorbing disk is a seminorm

rdf:langString 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 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 Minkowski function is always non-negative (meaning ) and is a real number if and only if is not empty. 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. In functional analysis, is usually assumed to have properties (such as being absorbing in for instance) that will guarantee that for every this set is not empty precisely because this results in being real-valued. Moreover, is also often assumed to have more properties, such as being an absorbing disk in since these properties guarantee that will be a (real-valued) seminorm on In fact, every seminorm on is equal to the Minkowski functional 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

rdf:langString Assume that is an absorbing subset of It is shown that: # If is convex then is subadditive. # If is balanced then is absolutely homogeneous; that is, for all scalars

rdf:langString If is a convex, balanced, and absorbing subset of a real or complex vector space then is a seminorm on

rdf:langString Let be any function. The following statements are equivalent: Strict positive homogeneity: for all and all real * This statement is equivalent to: for all and all positive real is a Minkowski functional; that is, there exists a subset such that where where Moreover, if never takes on the value then this list may be extended to include: Positive homogeneity/Nonnegative homogeneity: for all and all real

rdf:langString Let be any function and be any subset. The following statements are equivalent: is positive homogeneous, and is the Minkowski functional of , contains the origin, and is star-shaped at the origin. * The set is star-shaped at the origin if and only if whenever and A set that is star-shaped at the origin is sometimes called a .

rdf:langString Suppose that is a topological vector space over the real or complex numbers. Then the non-empty open convex subsets of are exactly those sets that are of the form for some and some positive continuous sublinear function on

rdf:langString Let be a subset of a real or complex vector space Then is a seminorm on if and only if all of the following conditions hold: ; is convex; * It suffices for to be convex. for all unit scalars * This condition is satisfied if is balanced or more generally if for all unit scalars in which case and both and will be convex, balanced, and absorbing subsets of Conversely, if is a seminorm on then the set satisfies all three of the above conditions and also moreover, is necessarily convex, balanced, absorbing, and satisfies

rdf:langString If is an absorbing disk in a vector space then the Minkowski functional of which is the map defined by is a seminorm on Moreover,

rdf:langString Suppose that is a subset of a real or complex vector space Strict positive homogeneity: for all and all real * Positive/Nonnegative homogeneity: is nonnegative homogeneous if and only if is real-valued. Real-values: is the set of all points on which is real valued. So is real-valued if and only if in which case * Value at : if and only if if and only if * Null space: If then if and only if if and only if there exists a divergent sequence of positive real numbers such that for all Moreover, Comparison to a constant: If then for any this can be restated as: If then * Thus if then where the set on the right hand side denotes and not its subset If then these sets are equal if and only if contains * In particular, if then but importantly, the converse is not necessarily true. Gauge comparison: For any subset thus if and only if * The set satisfies so replacing with will not change the resulting Minkowski functional. The same is true of and of * If then and has the particularly nice property that if is real then if and only if or Moreover, if is real then if and only if Subadditive/Triangle inequality: is subadditive if and only if is convex. If is convex then so are both and and moreover, is subadditive. Scaling the set: If is a scalar then for all Thus if is real then Absolute homogeneity: for all and all unit length scalars if and only if for all unit length scalars in which case for all and all scalars If in addition is also real-valued then this holds for scalars . * for all unit length if and only if for all unit length * for all unit scalars if and only if for all unit scalars if this is the case then for all unit scalars * is symmetric if and only if which happens if and only if * The Minkowski functional of any balanced set is a balanced function. Absorbing: If is convex balanced and if then is absorbing in * If a set is absorbing in and then is absorbing in * If is convex and then in which case Restriction to a vector subspace: If is a vector subspace of and if denotes the Minkowski functional of on then where denotes the restriction of to