Facial structure of convex sets 22–25 May 2017, Vancouver SIAM Conference on Optimization Vera Roshchina RMIT University and Federation University Australia Joint work with Tian Sang (RMIT University) and David Yost (Federation University Australia)
Faces of polytopes Recall that polytopes are convex hulls of finitely many points. The boundaries of these polytopes consist of convex polygons, which are in turn bounded by one-dimensional edges. We can also clearly distinguish the vertices (or extreme points). All these objects of different dimensions are faces of these polytopes. We can also define faces of general convex sets. 1/21
Definition of a face A face F of a convex set C ⊂ R n is a convex subset of C such that for any x ∈ F and for any line segment [ a, b ] ⊂ C such that x ∈ ( a, b ), we have a, b ∈ F . We write F ⊳ C for F face of C . Not a face Not a face 2/21
More details A face F of a convex set C ⊂ R n is a convex subset F of C such that for any point x ∈ F and for any line segment [ a, b ] ⊂ C such that x ∈ ( a, b ), we have a, b ∈ F . The two 'apices', all singletons Every singleton on the in the middle and every 'edge' boundary is a face; on the boundary are faces. the disk is a face. Note that C ⊳ C , ∅ ⊳ C . 3/21
A classic example The intersection of a sup- porting hyperplane with the set is always a face. Such faces are called exposed . Unlike in the polyhedral case, faces of general con- vex sets are not necessarily exposed. The convex hull of a torus [Rockafellar, Convex Analysis, 1970] has unexposed zero-dimensional faces (dashed line). 4/21
Dimensions of faces The dimension of a convex set is the dimension of its affine hull (the smallest affine subspace, such as point, line, plane, etc. that contains the set). Hence every face has a dimension. dim = 0 dim = 1 dim = 2 The two 'apices', all singletons Every singleton on the Every vertex, edge, Every singleton on the in the middle and every 'edge' boundary is a face; and all four two-dimensional boundary is a face on the boundary are faces. the disk is a face. facets are faces Recall that the set itself is a face (of dimension 3). 5/21
A question about dimensions Some convex sets have large gaps between the dimensions of their faces, for instance, the cone S n + of positive semidefinite n × n matrices has faces of dimensions k ( k + 1) / 2 for k ∈ { 0 , 1 , . . . , n } . A natural question is what are the possible patterns for the di- mensions of faces of closed convex sets? (0,3) (0,1,2,3) (0,2,3) (0,1,3) 6/21
A question about dimensions The empty set has the empty face only, we denote this by (). On the real line we have the empty set, the singletons and (pos- sibly unbounded) line segments. (0) (0,1) (1) On the plane the full dimensional possibilities are exhausted by circle, triangle, half space and the whole space. (0,1,2) (1,2) (2) (0,2) half the whole space space We have 8 = 2 3 sequences: () , (0) , (1) , (2) , (0 , 1) , (0 , 2) , (1 , 2) , (0 , 1 , 2). 7/21
Three dimensions The possibilities for three dimensional compact convex sets are exhausted by our examples: (0,3) (0,1,2,3) (0,2,3) (0,1,3) The remaining patterns are obtained as direct products of the two dimensional examples with the real line: (2) → (3), (0 , 2) → (1 , 3), (0 , 1 , 2) → (1 , 2 , 3) and (1 , 2) → (2 , 3). We have covered all increasing sequences of nonnegative integers that end in “3”: (3) , (0 , 3) , (1 , 3) , (2 , 3) , (0 , 1 , 3) , (0 , 2 , 3) , (0 , 1 , 2 , 3) . 8/21
All dimensional patterns are possible We can write down all potential dimensional patterns of the faces as an increasing sequence of nonnegative numbers ( d 0 , d 1 , d 2 , . . . , d k ). Theorem 1. (R, Sang, Yost) For any increasing sequence of nonnegative integers d = ( d 0 , d 1 , d 2 , . . . , d k ) there exists a closed convex set in d k -dimensional space such that the vector ( d 0 , d 1 , . . . , d k ) describes the pattern of facial di- mensions for this set. [R, Sang, Yost, Compact convex sets with prescribed facial dimensions, 2016] . 9/21
My collaborators David Yost Tian Sang Photo taken during the MATRIX research program on approxi- mation and optimisation in July 2016. 10/21
Minkowski sum Recall the definition of Minkowski sum : for C 1 , C 2 ⊆ R n we have C 1 + C 2 := { z = x + y | x ∈ C 1 , y ∈ C 2 } . The shape of the Minkowski sum can be visualised via ‘dragging’ one set over the boundary of the other one. + 0 11/21
A technical result Lemma 1. Let P, Q ⊂ R n be nonempty convex compact sets, and let C = P + Q . Then every face of C is the Minkowski sum of faces of P and Q . More precisely, ∀ F ⊳ C ∃ F P ⊳ P, F Q ⊳ Q such that F = F P + F Q . Minkowski sum of two orthogonal disks 12/21
Minkowski sum with the Euclidean ball Recall that for compact P, Q ⊂ R n and C = P + Q we have ∀ F ⊳ C ∃ F P ⊳ P, F Q ⊳ Q such that F = F P + F Q . When one of the sets, say, P , is the Euclidean ball, the sum can only have faces of the same dimension as the faces of Q , and the full-dimensional face. Sum of the Euclidean ball with a line segment and with a square. 13/21
Proof of the main result Recall that we want to prove that every finite in- creasing pattern of integers (1,3) (0,2) ( d 0 , d 1 , . . . , d k ) can be re- alised. We will prove that every such sequence that starts with zero, i.e. d 0 = 0, can be realised by a compact convex set. The result for the general closed convex sets is then straightfor- ward by taking products with subspaces of the required dimen- sion. 14/21
Proof of the main result: induction We use induction in the last number d k of the increasing sequence ( d 0 = 0 , d 1 , . . . , d k ) to demonstrate the result. Our base is the lower dimensional examples. Our assumption is that for any finite increasing sequence ( d 0 , . . . , d k ) of nonnegative numbers with d 0 = 0 and d k ≤ m we can con- struct a compact convex set in R d k with the corresponding facial pattern. We will show that for any sequence (increasing, starting from zero) ending in d k = m + 1 we can construct a compact convex set that realises this sequence. 15/21
Proof of the main result: reduction Our sequence: d = ( d 0 , d 1 , d 2 , . . . , d k ), d k = m + 1. Consider the truncated sequence: d ′ = ( d 0 , d 1 , d 2 , . . . , d k − 1 ). Since d k = m +1, we have d k − 1 ≤ m , and by inductive assumption there is a compact convex Q ′ ⊂ R d k − 1 that realises d ′ in R d k − 1 . Let Q := Q ′ × { 0 m +1 − l } ⊂ R m +1 . Now Q realises d ′ as well. 16/21
Proof of the main result: Minkowski sum We have Q ⊂ R m +1 that realises d ′ = ( d 0 , . . . , d k − 1 ). We claim that the dimensions of faces of C = Q + B (where B ⊂ R d k = R m +1 is the Euclidean ball) realise the full sequence d = ( d 0 , . . . , d k , d k +1 ). For this we need to show that the only dimension added is d k , and none of the original facial dimensions of Q are lost . The only faces of C are sums of faces of B and Q , hence they are either translations of faces of Q or of dimension d k . The plane x d k = 1 exposes the face of C that is an affine trans- lation of Q , hence, all original dimensions are present in C . 17/21
Some final remarks Observe that taking the union of faces of a given dimension of a polytope (or a polyhedral set) results in a set of dimension that coincides with the dimension of the faces. This is not the case for general setting: for instance, the dimen- sion of the union of all extreme points of a Euclidean ball in R n is n − 1. 18/21
Spherical gasket and Sierpinski triangles Consider their convex hulls. 19/21
An ugly picture of a beautiful set 20/21
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