FACES OF CONVEX SETS VERA ROSHCHINA Abstract. We remind the basic definitions of faces of convex sets and their basic properties. For more details see the classic references [1, 2] and [4] for polytopes. 1. Faces of convex sets Polytopes are convex hulls of finitely many points, and a polyhedral set is an intersection of finitely many half-spaces. Any polytope is a polyhedron, whilst a polyhedron may be unbounded. Platonic solids are polytopes (see Fig. 1). The boundaries of these sets consist of convex polygons Figure 1. Platonic solids: notice that their boundary consists of vertices, edges and two-dimensional facets. (called facets), which are in turn bounded by one-dimensional edges. We can also clearly distinguish the zero-dimensional vertices (or extreme points). All these objects of different dimensions are faces of these polytopes. Faces of convex sets generalise the notion of vertices, edges and facets of polytopes. Definition 1 (Face). Let C R n be a convex set. A convex set F C is called a face of C if for every x F and every y, z C such that x (y, z), we have y, z F. In other words, a face F is such a convex subset of a given convex set C that every line segment in C that intersects F at some point in its interior has to be included in F. The set C is its own face. Some examples of faces of convex sets are shown in Fig. 2. Every singleton on the boundary is a face Every vertex, edge, and all four two-dimensional facets are faces Every singleton on the boundary is a face; the disk is a face. The two 'apices', all singletons in the middle and every 'edge' on the boundary are faces. Figure 2. Various face arrangements of three dimensional convex sets: the only proper faces of a unit ball are extreme points; the proper faces of a tetrahedron come in all possible dimensions; half a unit ball has proper faces of dimensions 0 and 2 only; an intersection of two cones has proper faces of dimensions 0 and 1 only. The fact that F is a face of C is usually denoted by F C or F C. A face F C such that F C is called proper. An empty set is also a face of C. 1
2 VERA ROSHCHINA Lemma 1. Let C R n be a convex set, and suppose that F C. Let E F. Then E F if and only if E C. Proof. It follows from the definition that if the set E is a face of C, then it is also a face of F. Conversely, suppose that E is a face of F, and let y and z be in C be such that (y, z) E. Since E F, we have (y, z) F. Hence y, z F since F is a face of C. But then we also have y, z E, since E is a face of F. Zero dimensional faces are called extreme points. In other words, an extreme point of a convex set is a point which does not lie in any open line segment joining two points of the set. The set of all such points in a given convex set C is denoted by ext C. Before we talk about faces of different dimensions, let us define the dimension of a convex set. 2. The dimension of a convex set An affine subspace is a subset V of R n that can be represented as V = L + p, where L is a linear subspace of R n, and p R n. The linear subspace in this representation is unique, and the point p can be chosen arbitrarily from the set V. Affine subspaces are sometimes called flats. Note that affine subspaces are convex sets. The dimension of an affine subspace V = L + p is by definition the dimension of the linear subspace L. Observe that any hyperplane H(h, x 0 ) = {x R n x x 0, h = 0} is an affine subspace, H(h, x 0 ) = L + x 0, where L = {x x, h = 0}. Any affine subspace V R n, V R n is the intersection of a finite number hyperplanes (note that the whole space is the trivial hyperplane H = R n = {x x, 0 = 0}). If we choose a basis {h 1,..., h k } of L and consider the family of hyperplanes then L = H 1 H 2 H k. H i = {x R n x x 0, h i = 0} i {1,..., k}, Definition 2. The affine hull of a convex set C is the set of all finite affine combinations of its elements such that their coefficients sum to one, i.e. { p } p aff C = λ i x i λ i = 1, x i C. Just as the convex hull generalises the idea of the line segment connecting two points, the affine hull generalises the notion of a line drawn through two points. It is not difficult to observe that the affine hull of a set S R n is an affine subspace: indeed, choose an arbitrary point x 0 S. Then aff C = x 0 + span(s x 0 ). It can also be shown that the affine hull of the set S is the intersection of all affine subspaces that contain S, and hence is the smallest affine subspace that contains S. Definition 3 (Dimension of a convex set). The dimension of a convex set C is the dimension of its affine hull. Just as the affine hull is defined similarly to the linear span, it is convenient to talk about affine independence which is analogous to linear independence. An affine hull V = L + p of k + 1 points can have dimension at most k. Indeed, every point in V is represented as x = α k x k with α k = 1. Alternatively every point y in the linear subspace L can be represented as ( ) y = x x 0 = α k x k x 0 = α k x k α k x 0 = α k (x k x 0 ),
FACES OF CONVEX SETS 3 so that the vectors (x i x 0 ) span the linear subspace L. There are k such vectors, hence, the dimension of L is at most k. Definition 4. We say that the points x 1,..., x k are affinely independent if their affine hull has full dimension (which is k 1). The above definition is equivalent to saying that none of the points in the affinely independent system is an affine combination of the remaining points. Proposition 1. A finite collection of points x 0,..., x k is affinely independent if and only if the collection of points v i = x i x 0, i = 1,..., k is linearly independent. Proof. Hint: represent the affine hull as x 0 + L, where L = span(v 1,..., v k ). Example 1. Observe that the k-simplex k introduced earlier is the convex hull of k affinely independent points. Consider the representation { } k := x R k : x i = 1, x i 0 i = 1,..., k, then the affine hull of this set is the hyperplane { aff k = x R k : } x i = 1. 3. Supporting hyperplanes and exposed faces Definition 5 (Supporting hyperplane). A hyperplane is said to support a convex closed set C if the set C lies in its entirety in one of the affine half spaces defined by this hyperplane. The supporting hyperplane is proper if the set C does not lie in its entirety on this hyperplane. Theorem 1. If C R n is a nonempty closed convex set, then C is the intersection of all the closed half-spaces containing it. Proof. Denote by D the intersection of all half-spaces containing C. It is clear that D is a closed, convex set containing C, so it remains to show that D C. If this is not true, then there exists a point x 0 D that does not lie in C. Applying the separation theorem to x 0 and C, we find a hyperplane that separates x 0 from C, and this contradicts the definition of D. The result above provides an external characterization of closed convex sets as intersections of closed half-spaces. In contrast, the convex hull operation generates a convex set by enlargement, from inside. The facial structure of convex sets is infinitely more complex than the facial structure of polytopes and polyhedral sets. One of the core differences is highlighted by the fact that faces of convex sets may not be exposed. Definition 6 (Exposed face). We say that a face F of a closed convex set C is exposed if there exists a supporting hyperplane H to the set C such that F = C H. A classic example of a convex set that has unexposed faces comes from [2]: consider a torus and take its convex hull. The relative boundary of the new two dimensional face is the union of unexposed faces (see Fig. 3). Another example is a convex hull of a closed unit ball and a disjoint point. Observe that the definition of an exposed face is consistent in the sense that the intersection of a supporting hyperplane with a set is a face of that set. Proposition 2. An intersection of a closed convex set C with a supporting hyperplane is an exposed face of C.
4 VERA ROSHCHINA Figure 3. Convex hull of a torus is not facially exposed (the dashed line shows the set of non exposed extreme points). unexposed face Figure 4. An example of a two dimensional set and a three dimensional cone that have an unexposed face. Proof. Let F be a nonempty intersection of the closed convex set C with a supporting hyperplane so that and H = {x s, x = d}, s, x = d x F (1) s, x d x C. First of all, observe that F is convex. Let x F, x = (1 λ)y + λz where y, z C, λ (0, 1). We have d = x, s = (1 λ) y, s + λ z, s. In view of (1) we can only have y, s = z, s = d, and hence y, z F. This shows that F is a face of C. It is not difficult to observe that the intersection of any number of faces is still a face (which can be empty). A similar statement applies to exposed faces. Definition 7 (Conjugate face). For a closed convex set C R n and F C define the conjugate face F as F = {u C u, x = sup u, y x F }. y C Observe that for a closed convex cone K and any u K we have sup u, y = 0 y K ( 0 follows from duality, and from the fact that K is a cone the value can only be 0 or + ). Hence, for a closed convex cone K the above definition results in F = {u K u, x = 0 x F }. For a compact convex set C such that 0 int C the definition of a dual face is equivalent to F = {u C u, x = 1 x F }. Note that conjugate faces are always exposed.
FACES OF CONVEX SETS 5 4. Lifting and homogenisation There is a duality relation between polars of compact convex sets and the polars of their homogenisations that we explore in this section. The following statement relates the polar sets of C R n and K = cone({1} C) R n+1 (see Fig. 5) x 0 C K C K x 1 Figure 5. Homogenisation of polars Proposition 3. Let C R n be a compact convex set such that 0 n int C, and let K = cone({1} C). Then Proof. From the definition of a polar cone K = cone{{ 1} C }. K = {(y 0, ȳ) sup α(y 0 + x, ȳ ) 0} α 0, = {(y 0, ȳ) sup α>0, α(y 0 + x, ȳ ) 0} = {(y 0, ȳ) sup(y 0 + x, ȳ ) 0} = {(y 0, ȳ) sup x, ȳ y 0 }. Observe that since 0 n int C, for every y R n we have sup x C x, y > 0, hence, K = {α( 1, ȳ), α 0 α sup x, ȳ α} = {0 n+1 } {α( 1, ȳ), α > 0 sup x, ȳ 1} = {0 n+1 } {α( 1, ȳ), α > 0 ȳ C } = cone{{ 1} C }. A direct consequence of homogenisation is that facial structure of a pointed cone can be studied via its sections.
6 VERA ROSHCHINA References [1] Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. Abridged version of ıt Convex analysis and minimization algorithms. I and II. [2] R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970. [3] Rolf Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition, 2014. [4] Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.