Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional or gauge function is a function that recovers a notion of distance on a linear space. If formula_1 is a subset of a real or complex vector space formula_2 then the or of formula_1 is defined to be the function formula_4 valued in the extended real numbers, defined by formula_5 where the infimum of the empty set is defined to be positive infinity formula_6 (which is a real number so that formula_7 would then be real-valued). The Minkowski function is always non-negative (meaning formula_8) and  formula_7 is a real number if and only if formula_10 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, formula_1 is usually assumed to have properties (such as being absorbing in formula_2 for instance) that will guarantee that for every formula_13 this set formula_14 is not empty precisely because this results in formula_15 being real-valued. Moreover, formula_1 is also often assumed to have more properties, such as being an absorbing disk in formula_2 since these properties  guarantee that formula_15 will be a (real-valued) seminorm on formula_19 In fact, every seminorm formula_20 on formula_21 is equal to the Minkowski functional of any subset formula_1 of formula_21 satisfying formula_24 (where all three of these sets are necessarily absorbing in formula_21 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 formula_21 into certain properties of a function on formula_19 Minkowski functionals have been applied to describe phase behavior of nanoconfined fluids. Definition. Let formula_1 be a subset of a real or complex vector space formula_19 Define the of formula_1 or the associated with or induced by formula_1 as being the function formula_4 valued in the extended real numbers, defined by formula_33 where recall that  the infimum of the empty set is formula_6 (that is, formula_35). Here, formula_10 is shorthand for formula_37 For any formula_13 formula_39 if and only if formula_10 is not empty. The arithmetic operations on formula_41 can be extended to operate on formula_42 where formula_43 for all non-zero real formula_44 The products formula_45 and formula_46 remain undefined. Some conditions making a gauge real-valued In the field of convex analysis, the map formula_15 taking on the value of formula_6 is not necessarily an issue. However, in functional analysis formula_15 is almost always real-valued (that is, to never take on the  value of formula_6), which happens if and only if the set formula_10 is non-empty for every formula_52 In order for formula_15 to be real-valued, it suffices for the origin of formula_21 to belong to the or of formula_1 in formula_19 If formula_1 is absorbing in formula_2 where recall that this implies that formula_59 then the origin belongs to the algebraic interior of formula_1 in formula_21 and thus formula_15 is real-valued. Characterizations of when formula_15 is real-valued are given below. Motivating examples. Example 1 Consider a normed vector space formula_64 with the norm formula_65 and let formula_66  be the unit ball in formula_19 Then for every formula_13 formula_69 Thus the Minkowski functional formula_70 is just the norm on formula_19 Example 2 Let formula_21 be a vector space without topology with underlying scalar field formula_73 Let formula_74 be any linear functional on formula_21 (not necessarily continuous). Fix formula_76 Let formula_1 be the set formula_78 and let formula_15 be the Minkowski functional of formula_80 Then formula_81 The function formula_15 has the following properties: Therefore, formula_15 is a seminorm on formula_2 with an induced topology. This 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, formula_89 need not imply formula_90 In the above example, one can take a nonzero formula_91 from the kernel of formula_92 Consequently, the resulting topology need not be Hausdorff. Common conditions guaranteeing gauges are seminorms. To guarantee that formula_93 it will henceforth be assumed that formula_94 In order for formula_15 to be a seminorm, it suffices for formula_1 to be a disk (that is, convex and  balanced) and absorbing in formula_2 which are the most common assumption placed on formula_80 .</math> More generally, if formula_1 is convex and the origin belongs to the algebraic interior of formula_100 then formula_15 is a nonnegative sublinear functional on formula_2 which implies in particular that it is subadditive and positive homogeneous. If formula_1 is absorbing in formula_21 then formula_105 is positive homogeneous, meaning that formula_106 for all real formula_107 where formula_108 If formula_109 is a nonnegative real-valued function on formula_21 that is positive homogeneous, then the sets formula_111 and formula_112 satisfy formula_113 and formula_114 if in addition formula_109  is absolutely homogeneous then both formula_116 and formula_117 are balanced. Gauges of absorbing disks. Arguably the most common requirements placed on a set formula_1 to guarantee that formula_15 is a seminorm are that formula_1 be an absorbing disk in formula_19 Due to how common these assumptions are, the properties of a Minkowski functional formula_15 when formula_1 is an absorbing disk will now be investigated. Since all of the results mentioned above made few (if any) assumptions on formula_100 they can be applied in this special case. Convexity and subadditivity A simple geometric argument that shows convexity of formula_1 implies subadditivity  is as follows. Suppose for the moment that formula_126 Then for all formula_127 formula_128 Since formula_1 is convex and formula_130 formula_131 is also convex. Therefore, formula_132 By definition of the Minkowski functional formula_133 formula_134 But the left hand side is formula_135 so that formula_136 Since formula_137 was arbitrary, it follows that formula_138 which is the desired inequality. The general case formula_139 is obtained after the obvious modification. Convexity of formula_100 together with the initial assumption that the set formula_10 is nonempty, implies that formula_1 is absorbing. Balancedness and absolute homogeneity Notice that formula_1 being balanced implies that formula_144 Therefore formula_145  for any scalar formula_146</li> </ul> Topological properties. Assume that formula_21 is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let formula_1 be an absorbing disk in formula_19 Then formula_150 where formula_151 is the topological interior and formula_152 is the topological closure of formula_1 in formula_19 Importantly, it was assumed that formula_15 was continuous nor was it assumed that formula_1 had any topological properties. Moreover, the Minkowski functional formula_15 is continuous if and only if formula_1 is a neighborhood of the origin in formula_19 If formula_15 is continuous then formula_161 Minimal  requirements on the set. This section will investigate the most general case of the gauge of subset formula_1 of formula_19 The more common special case where formula_1 is assumed to be an absorbing disk in formula_21 was discussed above. Properties. All results in this section may be applied to the case where formula_1 is an absorbing disk. Throughout, formula_1 is any subset of formula_19 The proofs of these basic properties are straightforward exercises so only the proofs of the most important statements are given. The proof that a convex subset formula_169 that satisfies formula_170 is necessarily absorbing in formula_21 is  straightforward and can be found in the article on absorbing sets. For any real formula_172 formula_173 so that taking the infimum of both sides shows that formula_174 This proves that Minkowski functionals are strictly positive homogeneous. For formula_175 to be well-defined, it is necessary and sufficient that formula_176 thus formula_177 for all formula_178 and all real formula_179 if and only if formula_15 is real-valued. The hypothesis of statement (7) allows us to conclude that formula_181 for all formula_178 and all scalars formula_183 satisfying formula_184 Every scalar formula_183 is of the form formula_186 for some real formula_187 where formula_188  and formula_189 is real if and only if formula_183 is real. The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of formula_133 and from the positive homogeneity of formula_15 when formula_15 is real-valued. formula_194 Examples. The following examples show that the containment formula_195 could be proper. Example: If formula_196 and formula_197 then formula_198 but formula_199 which shows that its possible for formula_200 to be a proper subset of formula_201 when formula_202 formula_194 The next example shows that the containment can be proper when formula_204 the example may be generalized  to any real formula_205 Assuming that formula_206 the following example is representative of how it happens that formula_178 satisfies formula_208 but formula_209 Example: Let formula_178 be non-zero and let formula_211 so that formula_212 and formula_213 From formula_214 it follows that formula_215 That formula_216 follows from observing that for every formula_127 formula_218 which contains formula_219 Thus formula_208 and formula_221 However, formula_222 so that formula_223 as desired. formula_194 Positive homogeneity characterizes Minkowski functionals. The next theorem shows that Minkowski functionals are those functions formula_225 that have a certain purely algebraic property that is used widely. This theorem can be extended to  characterize certain classes of formula_226-valued maps (for example, real-valued sublinear functions) in terms of Minkowski functionals. For instance, it can be used to describe how every real homogeneous function formula_227 (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property. If formula_228 holds for all formula_178 and real formula_230 then formula_231 so that formula_232 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. So assume that  formula_225 is a function such that formula_234 for all formula_178 and all real formula_230 and let formula_237 For all real formula_172 formula_239 so by taking formula_240 for instance, it follows that either formula_241 Let formula_52 It remains to show that formula_243 It will now be shown that if formula_244 or formula_245 then formula_246 so that in particular, it will follow that formula_247 So suppose that formula_244 or formula_249 in either case formula_250 for all real formula_251 Now if formula_244 then this implies that that formula_253 for all real formula_230 (since formula_255), which implies that formula_256 as desired. Similarly, if formula_245  then formula_258 for all real formula_172 which implies that formula_260 as desired. Thus, it will henceforth be assumed that formula_261 a positive real number and that formula_262 (importantly, however, the possibility that formula_7 is formula_264 or formula_6 has not yet been ruled out). Recall that just like formula_266 the function formula_15 satisfies formula_268 for all real formula_251 Since formula_270 formula_271 if and only if formula_272 so assume without loss of generality that formula_273 and it remains to show that formula_274 Since formula_275 formula_276 which implies that formula_216 (so in particular, formula_39 is guaranteed). It remains to show that formula_279 which  recall happens if and only if formula_280 So assume for the sake of contradiction that formula_281 and let formula_282 and formula_283 be such that formula_284 where note that formula_283 implies that formula_286 Then formula_287 formula_194 Characterizing Minkowski functionals that are seminorms. In this next theorem, which follows immediately from the statements above, formula_1 is assumed to be absorbing in formula_21 and instead, it is deduced that formula_291 is absorbing when formula_15 is a seminorm. It is also not assumed that formula_1 is balanced (which is a property that formula_1 is often required to have); in its place is the weaker  condition that formula_295 for all scalars formula_183 satisfying formula_184 The common requirement that formula_1 be convex is also weakened to only requiring that formula_291 be convex. Positive sublinear functions and Minkowski functionals. It may be shown that a real-valued subadditive function formula_227 on an arbitrary topological vector space formula_21 is continuous at the origin if and only if it is uniformly continuous, where if in addition formula_302 is nonnegative, then formula_302 is continuous if and only if formula_304 is an open neighborhood in formula_19 If formula_227 is subadditive and satisfies formula_307 then formula_302 is continuous if and only if  its absolute value formula_309 is continuous. A positive sublinear function is a nonnegative homogeneous function formula_310 that satisfies the triangle inequality. It follows immediately from the results below that for such a function formula_266 if formula_304 then formula_313 Given formula_314 the Minkowski functional formula_15 is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if formula_316 and formula_291 is convex. Correspondence between open convex sets and positive continuous sublinear functions Let formula_318 be an open convex subset of formula_19 If formula_320 then let formula_321 and otherwise let formula_322 be arbitrary . Let formula_323 be the Minkowski functional of formula_324 where this convex open neighborhood of the origin satisfies formula_325 Then formula_20 is a continuous sublinear function on formula_21 since formula_328 is convex, absorbing, and open (however, formula_20 is not necessarily a seminorm since it is not necessarily absolutely homogeneous). From the properties of Minkowski functionals, it follows that formula_330 so that formula_331 Since formula_332 this completes the proof. formula_194