While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension

Branko Grünbaum

Erik M. Alfsen

G Pataki

Günter M. Ziegler

JE Humphreys

Richard D. Hill

T Sang

Ulrich Eckhardt

English

Vera Roshchina

Tian Sang

David Yost

Crossref

Compact Convex Sets with Prescribed Facial Dimensions

Roshchina, Vera

Sang, Tian

Yost, David

Federation ResearchOnline

  Page 1 of 1 Federation University ResearchOnline https://researchonline.federation.edu.au Copyright Notice  This is the post-peer-review, pre-copyedit version of an article published in 2016 Matrix Annals.  The final authenticated version is available online at:  https://doi.org/10.1007/978-3-319-72299-3_7  Copyright © Springer International Publishing AG, part of Springer Nature 2018  Use of the Accepted Manuscript is subject to AM terms of use, which permit users to view, print, copy, download and text and data-mine the content, for the purposes of academic research, subject always to the full conditions of use. Under no circumstances may the AM be shared or distributed under a Creative Commons, or other form of open access license, nor may it be reformatted or enhanced.   CRICOS 00103D RTO 4909   See this record in Federation ResearchOnline at: http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/182630 Compact convex sets with prescribed facialdimensionsVera Roshchina, Tian Sang and David YostAbstract While faces of a polytope form a well structured lattice, in which facesof each possible dimension are present, this is not true for general compact convexsets. We address the question of what dimensional patterns are possible for the facesof general closed convex sets. We show that for any finite sequence of positive in-tegers there exist compact convex sets which only have extreme points and faceswith dimensions from this prescribed sequence. We also discuss another approachto dimensionality, considering the dimension of the union of all faces of the samedimension. We show that the questions arising from this approach are highly non-trivial and give examples of convex sets for which the sets of extreme points havefractal dimension.1 IntroductionIt is well known that faces of polyhedral sets have a well-defined structure (see [12,Chap. 2]). In particular, every face of a polyhedral set is a polyhedron, and thereare no ‘gaps’ in the dimensions of their faces. On the other hand, a simple refor-mulation of [4, Corollary 3.7] asserts that in the compact convex set of all positivesemidefinite n×n matrices with trace 1, every proper face has dimension k2−1 forsome k < n. Thus there are naturally occuring examples with serious gaps in thedimensions of their faces. For other descriptions of this phenomenon, see Theorem2.25 and the explanation that follows it in [10] (for the cone Sn+ of positive semidef-Vera RoshchinaSchool of Science, RMIT University, e-mail: vera.roshchina@rmit.edu.auTian SangSchool of Science, RMIT University, e-mail: s3556268@student.rmit.edu.auDavid YostCentre for Informatics and Applied Optimisation, Federation University, e-mail:d.yost@federation.edu.au12 Vera Roshchina, Tian Sang and David Yostinite n×n matrices), or [1, Theorem 5.36] (for the state space of a C∗-algebra). Thisraises the question, what are the possible patterns for the dimensions of faces ofcompact convex sets?Recall that a face F of a closed convex set C⊂Rn is a closed convex subset of Csuch 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 fact that F is a face of C is expressed as F CC.The difference between this definition and the definition of faces of polyhedralsets as intersections with supporting hyperplanes is due to the fact that for non-polyhedral convex sets faces are not necessarily exposed: it may happen that a facecannot be represented as the intersection of a supporting hyperplane with the set.Some classic examples are shown in Figs. 1 (see [8]) and 2 (see [7]).Fig. 1 Convex hull of a torus is not facially exposed (the dashed line shows the unexposed extremepoints).unexposed faceFig. 2 An example of a two dimensional set and a three dimensional cone that have an unexposedface.The dimension of a convex set is the dimension of its affine hull, same for theface. We refer the reader to the classic textbooks [5, 8]. We also would like to men-tion that some problems related to dimensions of convex sets were studied in theliterature. For instance, [2] focusses on the dimensions of convex sets coming fromoptimisation problems with inequality constraints, and [3] deals with the results re-lated to the dimensions of intersections of convex sets. However, we were unable toidentify references that would address the existence of convex sets with prescribedfacial dimensions.Compact convex sets with prescribed facial dimensions 3The total number of possible face patterns in n dimensional space is the car-dinality of the powerset of n elements. This is because every set contains zero-dimensional faces (because of the Krein-Milman theorem). We can write down facepatterns either as an increasing sequence of positive numbers (d1,d2, . . . ,dk), whichencode all possible dimensions of faces of positive dimension present in a set, or asa binary sequence (b1,b2, . . . ,bn), where bi = 1 if a face of dimension i is present inthe set, and bi = 0 otherwise. For example, the dimensional pattern of a tetrahedronis either (1,2,3) in the d-notation or (1,1,1) in the binary notation, and the patternof a closed Euclidean ball is either (n) or (0,0, . . . ,1), as it does not have any facesexcept for zero- and n-dimensional ones. We will use the first encoding style via anincreasing sequence of positive numbers in what follows.The easiest cases to classify are the ones that we can visualise, i.e. the convexcompact sets in zero- one-, two- and three-dimensional spaces. In dimension zerowe have singletons {x} for any real x with pattern (), in one-dimensional space thereis no freedom: the only fully dimensional convex compact sets are line segments,with the only possible pattern (1). On the plane the two-dimensional possibilities areexhausted by a circle and a triangle, with patterns (2) and (1,2) respectively (seeFig. 3). Therefore for the two dimensional case we have four possibilities: (), (1),(1,2) and (2), which coincides with the cardinality of the powerset of two: 22 = 4.Fig. 3 All possible face patterns of fully dimensional sets in two dimensional case are given by adisk and a triangle.In three dimensions the possibilities for fully dimensional sets are exhausted bythe unit ball (3), the tetrahedron (1,2,3), the unit ball intersected with a closed half-space (2,3), and the convex hull of a circle in the plane and two points on oppositesides of the plane (1,3) (see Fig. 4), together with the lower dimensional exampleswe have in total 23 = 8 possibilities.2 Main ResultWe show that all patterns of facial dimensions can be realised in a compact convexset.Theorem 1. For any increasing sequence of positive integers4 Vera Roshchina, Tian Sang and David YostFig. 4 All possible facial patterns for the three dimensional setsd = (d1,d2, . . . ,dk)there exists a compact convex set in dk-dimensional space such that the vector ddescribes the pattern of facial dimensions for this set.To prove this, we need the following technical lemma, which is surely known,but we were not able to identify it in the literature. We hence provide a short proofhere as well.Lemma 1. Let P,Q⊂Rn be nonempty convex compact sets, and let C =P+Q. Thenevery face of C is the Minkowski sum of faces of P and Q. More precisely,∀F CC ∃FP CP,FQ CQ such that F = FP +FQ.Proof. Let F be a nonempty face of C. We construct two setsFP := {x ∈ P |∃y ∈ Q,x+ y ∈ F}, FQ := {y ∈ Q |∃x ∈ P,x+ y ∈ F}.Both FP and FQ are nonempty since F is nonempty.First we show that F = FP +FQ. It is obvious that F ⊂ FP +FQ, and it remains toshow the reverse inclusion. For that, pick an arbitrary x ∈ FP, y ∈ FQ. We will nextshow that z = x+ y ∈ F .By the definition of FP and FQ there exist u∈ P and v∈Q such that x+v∈ F andy+u ∈ F . If x = u or y = v, there is nothing to prove, as in this case z = u+ v ∈ F .Otherwise, by the convexity of F we havez′ =x+ v2+y+u2∈ F.At the same time, notice that x+ y ∈ P+Q ⊂C; likewise, u+ v ∈ P+Q ⊂C, andz′ ∈ (x+ y,u+ v). Since F is a face of C, this yields z = x+ y ∈ F .It remains to show that both FP and FQ are faces of P and Q respectively. Firstnote that both are closed compact sets, and that FQ ⊂ Q and FP ⊂ P.Let x ∈ FP, and pick any interval [a,b]⊂ P such that x ∈ (a,b). By the definitionof C, for an arbitrary y ∈ FQ we have a+ y,b+ y ∈ C. At the same time, x+ y ∈FP + FQ = F and x + y ∈ (a+ y,b+ y). From F CC we have [a+ y,b+ y] ⊂ F ,hence, a+ y,b+ y ∈ F , and therefore a,b ∈ FP. This shows that FP is a face of P.The proof for FQ is identical.Compact convex sets with prescribed facial dimensions 5In the proof of Theorem 1 presented next we use an inductive argument to explic-itly construct a compact convex set with a given facial pattern from a lower dimen-sional example for a truncated sequence. The key observation is that the Minkowskisum of an arbitrary compact convex set with a unit ball does not generate faces ofany new dimensions (compared to the original set) other than possibly the fully di-mensional face that coincides with the sum, which follows directly from Lemma 1.We sketched the Minkowski sum of two simple compact convex sets with a Eu-clidean ball in Fig. 5 to illustrate this argument.Fig. 5 Minkowski sum of a line segment and a unit sphere (on the left hand side), and of a unitsquare and a sphere (on the right hand side).Proof (Proof of Theorem 1). We use induction on dk to demonstrate the result. Ourinduction base is lower dimensional examples discussed earlier. For all increasingsequences of positive numbers (d1, . . . ,dk) with dk ≤ 2 we have found the relevantexamples. They are realised by a point, line segment, disk and triangle.Assume that our assertion is proven for all sequences (d1, . . . ,dk) with dk ≤ m.We will show that the statement is true for dk = m + 1. Choose an arbitrary se-quence d = (d1, . . . ,dk), where dk = m+ 1. If d = (dk), the sequence is realisedby the Euclidean unit ball in Rm+1. If the sequence contains more than one num-ber, consider the truncated sequence d′ = (d1,d2, . . . ,dk−1). Since dk−1 < dk, wehave l := dk−1 ≤ m, and there exists a compact convex set Q ⊂ Rl that realisesthe sequence d′ in l = dk−1-dimensional space. We embed the set Q in the m+ 1-dimensional space by letting Q′ := Q×{0m+1−l}. Observe that since the definitionof the face is algebraic, the facial pattern of the set Q′ is identical to the one of Q.Let B be the unit ball in Rm+1. We letC := B+Q′and claim that d is the facial pattern of C.From Lemma 1 every face of C can be represented as the sum of faces of Q′ andB. Since the only faces of B are the set itself and the singletons on the boundary,the only possible dimensions of the faces of the set C can come from the sequence6 Vera Roshchina, Tian Sang and David Yost(d1, . . . ,dk). To show that no facial dimensions are lost, observe that if e denotesthe unit vector (0,0, . . . ,1) ∈ B, then the set {e}+Q′ is a face of C (hence all itsfaces are also faces of C). Indeed, for the hyperplane H = {x | 〈e,x〉 = xm+1 = 1}supports C (notice that for every x = q+ b ∈ C with q ∈ Q′ and b ∈ B we havexm+1 = 0+qm+1 ≤ 1), moreover,H ∩C = {q+b |q ∈ Q′,b ∈ B,qm+1 +bm+1 = 1}= {q+b |q ∈ Q′,b ∈ B,bm+1 = 1}= {e}+Q′.It is not difficult to observe (e.g., see [8, Section 18]) that any supporting hyperplaneslices off a face from a convex set, hence, F = {e}+Q′CC. This face is linearlyisomorphic to Q, and hence the facial structure of F conicides with the facial struc-ture of Q, giving all possible dimensions of faces from the sequence d′. The faceof the maximal dimension m+ 1 is given by the set C itself, as it has a nonemptyinterior (take any point from Q′ and sum it with an open ball).3 Fractal convex setsObserve that polytopes not only possess faces of all possible dimensions, but theirfaces are also arranged in a very regular fashion: the union of the edges of a polytopeis a one-dimensional set (here we refer to a general notion of Hausdorff dimension,rather than the dimension of the affine hull that is useful for convex sets), the unionof all two dimensional faces is two dimensional, and so on. More generally, theunion of all faces of a polytope of a given dimension is a set of the same dimen-sion. This is not the case for a more general setting: for instance, the dimension ofthe union of all extreme points of a Euclidean ball in Rn is n− 1, a stark contrastwith the polyhedral case. Hence it is natural to study the dimension of the unionsof equidimensional faces. The purpose of this section is to present some exampleswhich emerged from the discussions during the MATRIX program, namely non-trivial sets with fractal facial structure and hence noninteger dimensions of the saidunions; these form the foundation for our ongoing research on this topic.Some work on fractals and convexity has been done before (see the recentwork [11] and references therein), but we are not aware of any references study-ing the particular problems that we propose here. We focus on two examples ofconvex sets that are generated in a natural way by spherical fractals. The finite rootsystem and Coxeter system are fundamental concepts in Lie algebras, which is veryimportant in many branches of mathematics. Given a finite root system, there is anatural associated finite Coxeter group, which is the Weyl group. People in the fieldof geometric group theory consider finite Coxeter groups are well-studied and ex-plained in liberature, see [6]. Therefore, we are more interested in the behaviours ofinfinite Coxeter groups. One such fractal comes from a recent work [9] by one ofCompact convex sets with prescribed facial dimensions 7our co-authors (curiously from the study of infinite Coxeter groups), another one isconstructed via projecting the Sierpinski triangle onto the unit sphere.We first consider a fractal set on a sphere and then take its convex hull, hencegenerating a convex set. Our first example is constructed in a similar way to theApollonian gasket: we take the unit sphere and construct a tetrahedron whose edgestouch the sphere (see Fig. 6), then consider the intersection of the sphere with theFig. 6 Construction of the spherical gasket.tetrahedron. After that, we continue slicing off spherical caps in such a way thatthey are tangential to the existing slices (see Fig. 7). The resulting body is a sphericalFig. 7 Apollonian gasket on a sphere and Sierpinski triangles.fractal, which is also a convex set. If we now take its convex hull, the extreme pointsof this convex set would be exactly the points on the fractal set, with remainingproper faces disks that result from the sliced off spherical caps. Notice that thisstructure is somewhat similar to the compact convex set obtained as the intersectionof the cone of symmetric positive semidefinite matrices of dimension 3×3 with anaffine subspace defined by matrices with a constant trace8 Vera Roshchina, Tian Sang and David YostC := S3+∩{M | tr(M) = 1}.This set has dimension 5 however.Algebraically, this particular fractal set is generated by the infinite Coxeter groupwith following group presentation:G = 〈s1,s2,s3,s4 | (si)2 = (sis j)∞ = 1〉The fractal sets are generated by limit roots, see [9]. Limit roots exhibit peculiar ge-ometric behaviour. Even though Coxeter groups are generated by affine reflectionsacross hyperplanes, when we compute the roots of the group and project them downto a lower dimensional affine hyperplane, the set of limit roots behaves like a frac-tal set, giving self-similar patterns that cannot be obtained by reflecting across anyhyperplanes.This approach can be applied to constructing other spherical fractals. For in-stance, one can generalise the Sierpinski carpet by cutting out triangular pieces ofthe sphere in a similar fashion. The convex set obtained after taking the convex hullof this spherical fractal will have faces of all possible dimensions.The Hausdorff dimension of the union of the extreme points is non-integer in bothcases, and coincides with the dimension of the relevant two-dimensional objects.It would be interesting to study the conditions that can be imposed on the facialdimensions to define good or regular convex sets.Acknowledgements The ideas in this paper were motivated by the discussions that took placeduring a recent MATRIX program in approximation and optimisation held in July 2016. We aregrateful to the MATRIX team for the enjoyable and productive research stay. We would also liketo thank the two referees for their insightful corrections and remarks.References1. Erik M. Alfsen and Frederic W. Shultz. State spaces of operator algebras: Basic theory,orientations, and C∗-products. Mathematics: Theory & Applications. Birkhäuser Boston, Inc.,Boston, MA, 2001.2. Ulrich Eckhardt. Theorems on the dimension of convex sets. Linear Algebra and Appl.,12(1):63–76, 1975.3. Branko Grünbaum. The dimension of intersections of convex sets. Pacific J. Math., 12:197–202, 1962.4. Richard D. Hill and Steven R. Waters. On the cone of positive semidefinite matrices. LinearAlgebra Appl., 90:81–88, 1987.5. Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of convex analysis.Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. Abridged version of Convex anal-ysis and minimization algorithms I and II.6. James E. Humphreys. Reflection Groups and Coxeter Groups. Cambridge University Press,Cambridge, 1990.7. Gábor Pataki. On the connection of facially exposed and nice cones. J. Math. Anal. Appl.,400(1):211–221, 2013.Compact convex sets with prescribed facial dimensions 98. R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. PrincetonUniversity Press, Princeton, N.J., 1970.9. Tian Sang. Limit roots for some infinite Coxeter groups. Master’s Thesis, Department ofMathematics and Statistics, University of Melbourne, 2014.10. Levent Tunçel. Polyhedral and semidefinite programming methods in combinatorial optimiza-tion, volume 27 of Fields Institute Monographs. American Mathematical Society, Providence,RI; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2010.11. József Vass. On the Exact Convex Hull of IFS Fractals. arXiv:1502.03788v2, 2016.12. Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics.Springer-Verlag, New York, 1995.

Compact convex sets with prescribed facial dimensions

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Compact convex sets with prescribed facial dimensions

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