VADU 2018 Open Problem Session
VVADU 2018 Open Problem Session
Bui Thi Hoa ∗ , Scott B. Lindstrom † and Vera Roshchina ‡ * June 5, 2018
Abstract
We state the problems discussed in the open problem session atVariational Analysis Down Under (VADU2018) conference held inhonour of Prof. Asen Dontchev’s 70th birthday on 19–21 February2018 at Federation University Australia, https://sites.rmit.edu.au/asen/ . Contents -polytopes -linked? 23 Is FFS3 polytope decomposable? 34 Projections onto compact convex sets 35 Convergence of the continuous time Douglas-Rachford algo-rithm 46 Minimal distance problem 57 Demyanov-Ryabova conjecture 68 D¨urer’s conjecture 6 ∗ CIAO, Federation University Australia † School of Mathematical and Physical Sciences, University of Newcastle, Australia ‡ School of Mathematics and Statistics, UNSW and School of Science, RMIT University,Australia a r X i v : . [ m a t h . O C ] J un Existence of local calm selections
This problem was proposed by Asen Dontchev. All background material,including notation, history, etc. can be found in [13]. We are grateful toAsen for providing this description.
Theorem (Bartle-Graves (1952)).
Let X and Y be Banach spaces andlet f : X → Y be a function which is strictly differentiable at ¯ x and suchthat the derivative Df (¯ x ) is surjective. Then there exist a neighborhood V of f (¯ x ) and a constant γ > such that f − has a continuous selection s on V which is calm with constant γ ; that is, (cid:107) s ( y ) − ¯ x (cid:107) ≤ γ (cid:107) y − f (¯ x ) (cid:107) for every y ∈ V. When X and Y are finite dimensional, even Hilbert, the proof is easy.For Banach spaces, the proof is highly nontrivial. A generalization of theBartle-Graves theorem to set-valued mappings was obtained in [12].Here is the open problem: Conjecture.
Consider a function f : R n → R m which is Lipschitz con-tinuous around ¯ x and suppose that all matrices A in Clarke’s generalizedJacobian of f at ¯ x are surjective. Then f − has a continuous local selectionaround ¯ y for ¯ x which is calm at ¯ y = f (¯ x ) . If n = m the conjecture reduces to Clarke’s inverse function theorem. For m ≤ n , according to a theorem by Pourciau [19], under the same conditionthe function f is metrically regular. This last result was generalized recentlyto Banach spaces in [7]. -polytopes -linked? This problem was presented by Bui Thi Hoa.A graph G is k -linked if for any selection of k pairs of all distinct vertices Y := { ( s , t ) , . . . , ( s k , t k ) } , ( k ≥
1) there exist k disjoint paths, connectingthe k pairs of points in Y . If the graph of a polytope is k -linked we say thatthe polytope is also k -linked .Recall that a d -polytope is a d -dimensional polytope, i.e. the linear spanof the polytope is a d -dimensional space. The initial question is whether ornot every d -polytope is (cid:98) d/ (cid:99) -linked . And the negative answer was given byGallivan (see [22]) with a construction of a d -polytope which is not (cid:98) d +4) / (cid:99) -linked . It had been already proven that 4-polytopes and 5-polytopesare 2 -linked (see [6], [8]), meanwhile not all 8-polytopes are 4 -linked . Theremaining question is that if all the 6-polytopes are 3 -linked .2 Is FFS3 polytope decomposable?
This problem was suggested by David Yost, and communicated during theopen problem section by Scott Lindstrom and Vera Roshchina.A polytope is called decomposable [20] if it can be represented as Minkowskisum of dis-similar convex bodies. Two polytopes are similar if one can beobtained from the other by a dilation and a translation.David Yost in collaboration with Debra Briggs have classified all but one3-polytopes with up to 16 edges in terms of decomposability (manuscript inpreparation). The only remaining case is the (combinatorial) polytope FFS3with its graph shown in Fig. 1. It is conjectured that this polytope has noFigure 1: Graph of the polytope FFS3decomposable geometric realisation.All polytopes with up to 15 edges are classified in terms of their decom-posability [5], and the resolution of the decomposability question for FFS3polytope will settle the 16-edge case. However further case-by-case decom-posability classification of polyhedra with higher number of edges presents atedious challenge, and a more interesting question is developing an algorithmto check decomposability. We note that in an overwhelming number of casesindecomposability can be checked using combinatorial conditions from [20].
This problem was proposed by Andrew Eberhard.Let C and C be compact convex sets in a Hilbert space H . The con-jecture states that there always exists a point x ∈ H such that for each ofits projections p i onto C i , i ∈ { , } the relevant normals x − p and x − p define the hyperplanes that strongly expose the faces { p } and { p } of C and C respectively. 3ecall (see [15, Definition 8.27]) that a point x ∈ C is strongly exposedby a linear functional f if f ( x ) = sup x (cid:48) ∈ C f ( x (cid:48) ) and x k → x for all sequences { x k } ⊂ C such that lim f ( x k ) = sup x ∈ C f ( x ). This problem was proposed by Scott Lindstrom.For the feasibility problem of finding a point in the nonempty intersection A ∩ B (cid:54) = ∅ of proximal sets A and B , the Douglas-Rachford method for agiven starting point x generates a sequence x n ∈ T x n − := ( λ (2 P B − Id)(2 P A − Id) + (1 − λ )Id) x n − where P A , P B denote the usual projection operators for A, B respectively and λ ∈ (0 ,
1] is usually taken to be 1 /
2. When
A, B are also convex, the sequence( x n ) n ∈ N converges weakly to a fixed point of the method (see [18] and [1]). - - - - - - - - Figure 2: The flowfield (1) with a circle/line (left) and ellipse/line (right).Images courtesy of Veit Elser.For the nonconvex case where A is a circle and B a line, Borwein andSims [4] considered the “continuous time” version of the algorithm—whoseflow field is shown at left in Figure 5 and corresponds to the solution of thedifferential equation given by dxdt = T ( x ) when λ → + (1)4igure 3: Behaviour of Douglas-Rachford method with an ellipse and linevaries from the case of a circle and line.—as a means to approaching the question of convergence in the usual case of λ = 1 / This problem was proposed by Alex Kruger.Given a finite set of points a , . . . , a m ∈ X , where X is an Euclideanspace, find the solution to the problemmin x ∈ X max i ∈{ ,...,m } (cid:107) a i − x (cid:107) . (2)The problem has a unique solution for which x is the centre of the minimalEuclidean sphere that contains all points. However it is unclear whetherthere exists a neat way to write this explicitly.This is a particular case of a more general problem. The space X can bean arbitrary normed linear or even a metric space. In the latter case, thenorm of the difference in (2) should be replaced by the distance. Instead ofthe maximum in (2), it could be an arbitrary norm in R m .5 Demyanov-Ryabova conjecture
This problem was communicated by Vera Roshchina.The problem was originally stated in [11, Conjecture 1]. Recently twodifferent special cases were confirmed in [9, 23]. During the preparation ofthis file a counterexample was found [21].Given a finite family Ω of convex polytopes in R n , for each unit vector g ∈ S n − we construct a new polytope as the convex hull of all support facesof all polytopes in the family Ω, i.e. we define the function C ( g ) := conv { Arg max x ∈ P (cid:104) x, g (cid:105) | P ∈ Ω } . Collecting all such polytopes, we obtain a new finite family of polytopes, F (Ω) = { C ( g ) g ∈ S n − } . Now starting from a given finite collection of polytopes Ω we apply thistransformation infinitely obtaining a sequence Ω , Ω , Ω , . . . , where Ω i = F (Ω i − ), i ∈ N .The original Demyanov-Ryabova conjecture claimed that this sequenceeventually reaches a two-cycle, i.e. for a sufficiently large N we have Ω N +2 =Ω N . Since we now know that the conjecture is false, the question is tofind a characterisation of such collections of polytopes that yield two-cycles,extending and generalising the results of [9, 23]. This problem was communicated by Vera Roshchina.Albrecht D¨urer dedicated a nontrivial part of his career to laying outthe geometric foundations of drawing and perspective. His five centuries oldwork [14] is available online via Google books. The mathematical statementknown as D¨urer’s conjecture was motivated by this work and proposed in1975 by Shephard [24]. A net (or unfolding) of a 3-polytope is the processof cutting it along its edges, so that the resulting connected shape can beflattened (developed) into the plane [17]. It is not difficult to find examplesof polytopes for which certain cuts result in overlapping unfoldings, such asthe truncated tetrahedron shown in Fig. 4 (see [16]).The D¨urer’s conjecture is a claim that any polytope has a nonoverlap-ping net. A significant recent contribution in this direction is the work byMohammed Ghomi who showed that every polytope is combinatorially equiv-alent to an unfoldable one [16]. For more details we refer the reader to anoverview [17] by the same author. 6igure 4: Two different nets of the same truncated tetrahedron
Acknowledgements
We are grateful to Asen Dontchev, Andrew Eberhard, Alex Kruger and DavidYost for patiently clarifying the mathematical details of their open problemsto us.
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