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Support Functions of General Convex Sets Algebra univers. 49 (2003) 305–319 0002-5240/03/030305 – 15 DOI 10.1007/s00012-003-1719-2 c Birkh¨auser Verlag, Basel, 2003 Algebra Universalis Support functions of general convex sets Dug-Hwan Choi and Jonathan D. H. Smith Dedicated to the Memory of Gian-Carlo Rota 1. Introduction The concept of the support function of anon-emptycompact convex set was introduced by Minkowski at the end of the 19th century [3, pp. 106, 144, 231]. Since then it has played a vital rˆole in many of the applications of convexity, from optimization theory to the geometry of numbers. Support functions of non-empty compact convex subsets of a finite-dimensional Euclidean space Rd are character- ized as positively homogeneous convex real-valued functions on Rd,andnon-empty compact convex sets are determined uniquely by their support functions. Work of Romanowska and the second author [5], [7, §11] extended the concept of the support function from compact non-empty convex sets to general bounded non- empty convex sets, thus to convex sets that are not necessarily closed. The method was to find a suitable codomain Dd,replacing the codomain R of Minkowski’s support functions, so that non-empty bounded convex subsets of Rd are determined d uniquely by their Dd-valued support functions defined on R .Conditions were also found, analogous to Minkowski’s positive homogeneity and convexity, characterizing d the support functions amongst all the Dd-valuedfunctions on R .Thesewerethe so-called G-conditions of [5, Definition 4.11], [7, Definition 11.2]. The purpose of the current paper is to produce support functions (Definition 7.1) specifying general convex subsets ofafinite-dimensional Euclidean space Rd, essentially removing the restriction to the bounded, non-empty case that was im- ± posed in [5]. A codomain Dd is constructed so that arbitrary convex subsets of d ± d R are determined uniquely by their Dd -valued support functions defined on R . ± d These support functions are characterized amongst all Dd -valuedfunctions on R by satisfaction of a set of conditions called the G±-conditions (Section 6 below), analogues in turn of the G-conditions of [5] and of Minkowski’s positive homogeneity and convexity conditions. Presented by J. Kung. Received September 2, 1999; accepted in final form December 14, 2000. 2000 Mathematics Subject Classification: 52A20; 06F99. Key words and phrases: Support function, convex set, entropic algebra, mode, modal, barycen- tric algebra, semilattice, lexicographic order, ordinal product, ordinal sum, Plonka sum. 305 306 D.-H. ChoiandJ.D.H. Smith Algebra univers. The correspondences between convex sets and their support functions go fur- ther. Algebraic structure on R induces algebraic structure on the set of real-valued functions on Rd.Thesetofsupport functions of non-empty, compact convex sets should be closed under this algebraic structure, and should reflect comparable alge- braic structure on the set of non-empty compact convex subsets of Rd.Theusual linear algebraic structure on R is unsuitable here: for example, the negative of a convex function is no longer convex.Thekeyalgebraic structure on R for use in the context of support functions comprises convex combinations forming a barycentric algebra (see [4]) and the maximum operation forming a join semilattice, the convex combinations distributing over the join so that the two structures combine to form a modal in the sense of [4]. The support functions then form a submodal of the induced modal structure on the full set of functions, and the modal structure on the support functions reflects exactly the modal structure on the compact convex sets given by convex combinations and convex hulls of unions. (See Section 3 below for an outline, and [4, 3.7] for details.) Given this algebraic approach to Minkowski’s support functions, and the fact that the set of all convex subsets of Rd forms a modal under convex combinations and convex hulls of unions, it is then shown that ± the codomain Dd carries a modal structure (Proposition 5.1) such that support functions form a submodal (Corollary 8.6) of the induced modal of functions from d ± R to Dd ,thissubmodal being isomorphic to the modal (3.3) of all convex subsets of Rd (Theorem 8.7). 2. Algebraic preliminaries An algebra (A, Ω) of type τ :Ω→ N is entropic if each operation ω is a homo- morphism ωτ ω :(A , Ω) → (A, Ω) ; (a1,...,aωτ ) → a1 ···aωτ ω. (2.1) An algebra is idempotent if each singleton subset is a subalgebra. An algebra is a mode if it is idempotent and entropic. An algebra (A, +, Ω) is a modal if (A, +) is ajoinsemilattice, (A, Ω) is a mode, and for each operation ω in Ω, the distributive law a1 ···(ai + ai) ···aωτ ω = a1 ···ai ···aωτ ω + a1 ···ai ···aωτ ω (2.2) holds for 1 ≤ i ≤ ωτ. For a real number r,define a binary operation r on R by r : R2 → R ;(x, y) → x(1 − r)+yr. (2.3) Let I◦ =[0, 1]◦ denote the interior of the unit interval I.Let D =(R, +,I◦)(2.4) Vol. 49, 2003 Support functions of general convex sets 307 denote the algebra structure on R given by the join semilattice operation x + y = max{x, y} and the binary operations p of (2.3) for p in I◦.ThenD is a modal. Now let R±∞ = R ∪{−∞, ∞} be the ordered set of (doubly) extended reals.Foreach element r of the set R+ of strictly positive reals, a binary operation r will be defined on R±∞ to yield an algebra (R±∞, R+). Define xy1 = y.Forreal arguments x and y,define xyr using (2.3), so that (R, R+)isasubalgebra of (R±∞, R+). For {x, y} ⊆ R and p ∈ I◦,define xyp = if −∞ ∈ x, y then −∞ else ∞ (2.5) For {x, y} ⊆ R and q ∈ (1, ∞), define xyq = if x<∞ and y = ∞ then ∞ else −∞ (2.6) Thus {−∞} is also a subalgebra of (R±∞, R+), and a sink of (R±∞,I◦). Bearing in mind that (R, R+)isasubalgebra of (R±∞, R+), one may summarize the algebraic structure on the doubly extended reals by the tables I◦ −∞ R ∞ −∞ −∞ −∞ −∞ R −∞ R ∞ (2.7) ∞ −∞ ∞ ∞ and (1, ∞) −∞ R ∞ −∞ −∞ −∞ ∞ R −∞ R ∞ (2.8) ∞ −∞ −∞ −∞. 3. Extended real valued support functions For a finite dimension d,letRd denote the vector space Rd equipped with the Euclidean inner product Rd × Rd → R ;(x, y) → (x|y). (3.1) The vector space Rd also supports a mode structure (Rd, R+)oftypeR+ ×{2} given as the d-th power of the mode (R, R+)using(2.3). [The d-th power semilattice structure on Rd will not be needed, so that + will denote addition on Euclidean spaces Rd,including R1,while (R, +) is the semilattice of (2.4).] Note that the convex subsets of Rd are precisely thesubalgebras of (Rd,I◦). Let RdT denote the set of all convex subsets of Rd.Thissetbecomes a mode (RdT,I◦) under the complex products XYp = {xyp | x ∈ X, y ∈ Y }. (3.2) 308 D.-H. ChoiandJ.D.H. Smith Algebra univers. Furthermore, (RdT,+) is a join semilattice under the containment relation ⊆;for convex subsets X and Y of Rd,thejoinX + Y is the convex hull of X ∪ Y .One thus obtains a modal structure (RdT,+,I◦)(3.3) on the set of convex subsets of Rd [7, §7]. Definition 3.1. Consider the function H : RdT × Rd → R±∞ ;(X, y) → sup{(x|y) | x ∈ X}. (3.4) For a fixed convex subset X of Rd,the(extended real valued) support function of X is the function d ±∞ HX : R → R ; y → H(X, y). (3.5) For an element y of Rd,thesupporting hyperplane of X in the y-direction is d z ∈ R (x|y)=HX (y) . (3.6) Note that (3.6) will be empty if HX (y)isnotfinite. Also, note HX (0) = if X = ∅ then −∞ else 0(3.7) Thus the supporting hyperplane of ∅ in any direction is empty. For a non-empty convex subset of Rd,thesupporting hyperplane in the 0-direction is Rd. Proposition 3.2. For a convex subset X of Rd,theclosure X of X is given by d X = z ∈ R (z|y) ≤ HX (y) . (3.8) y∈Rd d Proof. For any element y of R ,onehas H∅(y)=−∞,sothat(3.8) holds for empty X.Nowsuppose that X is non-empty. For each element (x, y)ofX × Rd, one has (x|y) ≤ sup{(z|y) | z ∈ X} = HX (y). Thus X is contained in the closed set on the right hand side of (3.8). Conversely, suppose that t does not lie in X.Then 0doesnotlie in the translated closed convex set X − t.Itfollows that there is a vector y with 0 >HX−t(y)=sup{(x − t|y) | x ∈ X} =sup{(x|y) | x ∈ X}−(t|y), whence (t|y) > sup{(x|y) | x ∈ X}≥sup{(x|y) | x ∈ X} = HX (y)andt does not lie in the right hand side of (3.8). Afunction f : Rd → R±∞ is said to be positively homogeneous if f(0) ∈{0, −∞} and ∀r ∈ R+, ∀y ∈ Rd,f(0yr)=f(0)f(y)r. (3.9) d Proposition 3.3. For eachconvex subset X of R ,thesupport function HX is positively homogeneous. Vol. 49, 2003 Support functions of general convex sets 309 Proof. In view of (3.7), it suffices to verify HX (yr)=HX (0yr)=HX (0)HX (y)r (3.10) for all y in Rd and r in R+.IfX is empty, then all the terms of (3.10) are −∞, ±∞ + since {−∞} is a subalgebra of (R , R ). Otherwise, suppose that HX (y)=∞, so that HX (yr)=∞ also.
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