Csaba Vincze (2013)
University of Debrecen
In this section we consider two convex functions which are closely related to the geometry of convex sets. Because their applications are limited in this material most of the basic properties will be left as an excercise.
Definition A compact set in the coordinate space of dimension n is called a body if it is the closure of its interior.
Definition Let K be a convex body containing the origin in its interior. The Minkowski functional induced by Kis defined as
where t > 0. Minkowski spaces are finite dimensional real vector spaces equipped with a Minkowski functional.
It can be easily seen that the Minkowski functional is positively 1-homogeneous and subadditive: for any positive real number t
Subadditivity together with positively homogenity imply convexity as well. The symmetry of K with respect to the origin is equivalent to the absolute homogenity (or reversibility)
(cf. properties of a norm in the space). In the preamble to his fourth problem presented at the International Mathematical Congress in Paris (1900) Hilbert suggested the examination of geometries standing next to Euclidean one in the sense that they satisfy much of Euclidean's axioms except some (tipically one) of them. In the classical non-Euclidean geometry the axiom taking to fail is the famous parallel postulate. Another type of geometry standing next to Euclidean one is the geometry of normed spaces (Minkowski spaces). The crucial test is not the parallelism but the congruence via the group of linear isometries. In his pioneering work on the geometry of numbers Minkowski realized that the best way for the investigation of normed spaces (Minkowski spaces) is to consider the unit sphere or the unit ball. Conversely, if we have a compact convex subset K containing the origin in its interior then the functional 7.5 measures the length of vectors in an adequate way. Moreover the distance function d with respect to l can be also introduced in the usual way as lengths of difference vectors. Geometrically the length is a simple ratio in the sense that
where l* is an arbitrary function (norm) measuring the lenght of vectors and v* is the boundary point of K corresponding to the ray from the origin into the given direction v. The functional l was first defined by H. Minkowski to provide a method of obtaining a norm together with a topology in very general linear spaces.
Definition By a linear isometry with respect to the Minkowski functional l we mean a linear transformation (invertible linear map) preserving the Minkowskian lenght of vectors.
The study of isometries or distance-preserving mappings of a Minkowski space is greatly simplified by a celebrated theorem due to Mazur and Ulam  for normed spaces or Minkowski spaces with reversible Minkowski functionals. The theorem states that any surjective isometry can be written as the composition of an invertible linear map and a translation. The linear part is, of course, automatically a linear isometry of the space. Therefore a general Minkowski space does not admit isometries of many types onto itself. In general translations, the identity map and the central symmetry with respect to the origin are the only examples.
Remark The converse of the problem of isometries seems to be also important: how to find an invariant convex body under a given subgroup of invertible linear transformations? The problem will be discussed in chapter 10.
Definition Let S be a non-empty convex set. The support function of S is an extended real-valued function defined as
The domain of an extended real-valued function is the set of points where the function has finite values at.
Extended real valued functions admit the positive or negative infinity as values. The set of extended real numbers has an ordering extending the natural ordering on the set of reals. Therefore the sup-operator can not cause any confusion in the definition of the support function. To clarify the geometric meaning of the support functions associated with convex sets we need the notion of polar sets.
Definition Let L be a non-empty subset in the coordinate space of dimension n. The polar set L* is defined as
Remark In case of a singleton the polar set is just the space itself if the origin is the only element of L. Otherwise if p is the only element in L but different from the origin then the characteristic property of the elements in L* is equivalent to the inequality
which gives a closed half-space. It can be easily seen that the bounding hyperplane is orthogonal to p.
In the sense of Riesz representation theorem each linear functional in the dual space can be expressed as
where n is a uniquely determined vector belongig to f. Conversely the right hand side of the formula defines a linear functional. Using the identification between f and n the polar set L* can be identified with the set of all linear functionals having supremum at most one on L.
Theorem 7.3.1 Let K be a compact convex body containing the origin in its interior. The support function of K is equal to the Minkowski functional of the polar set K*.
Proof Since K is compact the domain of the support function is the coordinate space of dimension n. On the other hand the origin in the interior of K implies that the support function is a positive definite (positively 1-homogeneous and convex) function. The equivalence
is also obvious. Therefore the unit ball with respect to the support function is just the polar set of K. ▮
To justify the name "support" we present the following theorem.
Theorem 7.3.2 Let K be a compact convex set and consider a non-zero element v in the coordinate space of dimension n. The hyperplane H
supports K at the point where the supremum 7.7 is attained at.
Proof The hyperplane H bounds K because for any element p in K
On the other hand the point where the supremum 7.7 is attained at is a common point of H and K. ▮
Remark The distance of H from the origin is just
Keeping in mind Hilbert's motivations why to investigate the geometry of normed spaces (Minkowski spaces) we illustrate one more problem which is closely related to the basic notions of Euclidean geometry: orthogonality/normality , see also . It is very important to see that the way of measuring angles between vectors in a Minkowski space is a non-trivial problem. In what follows wesketch a way to introduce the notion of orthogonal vectors. Sometimes they are called normal to each other provided that the role of the vectors is symmetric.
Definition Taking two unit vectors v and w in a Minkowski plane we say that v is normal to w if the line passing through v into the direction w supports the unit disk at v.
Definition The Radon plane is such a Minkowski plane for which normality is symmetric. The unit circles of Radon planes are called Radon curves.
For the characterization of Radon curves and examples see the corresponding list of excercises.