Rigidity and polyhedral The American Institute of

The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Rigidity and .” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Mon Nov 26 14:43:28 2007

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Table of Contents A. Participant Contributions ...... 3 1. Alexandrov, Victor 2. Bezdek, Karoly 3. Bobenko, Alexander 4. Dolbilin, Nikolay 5. Fernandez, Silvia 6. Fillastre, Francois 7. Graver, Jack 8. Izmestiev, Ivan 9. Martin, Jeremy 10. Nevo, Eran 11. Panina, Gayane 12. Rote, Gunter 13. Sabitov, Idzhad 14. Schlenker, Jean-Marc 15. Schulze, Bernd 16. Servatius, Brigitte 17. Stachel, Hellmuth 18. Tabachnikov, Sergei 19. Tarasov, Alexey 20. Thurston, Dylan 21. Whiteley, Walter 22. Ye, Yinyu 3

Chapter A: Participant Contributions

A.1 Alexandrov, Victor In the moment my research interests are focused at the Strong Bellows Conjecture which reads as follows: every two polyhedra in Euclidean 3-space, such that one of them is obtained from the other by a continuous flex, are scissor congruent, that is, one of them can be cut into “small” simplices which, being moved independently one from another, give a partition of the other . A.2 Bezdek, Karoly Recall that the intersection of finitely many closed unit balls of E3 is called a ball- polyhedron (for more details see [blnp]). Also, it is natural to assume that removing any of the balls in question yields the intersection of the remaining balls to become a larger set. One can represent the boundary of a ball-polyhedron in E3 as the union of vertices, edges and faces defined in a rather natural way as follows. A boundary point is called a if it belongs to at least three of the closed balls defining the ball-polyhedron. A of the ball-polyhedron is the intersection of one of the generating closed balls with the boundary of the ball-polyhedron. Finally, if the intersection of two faces is non-empty, then it is the union of (possibly degenerate) circular arcs. The non-degenerate arcs are called edges of the ball-polyhedron. Obviously, if a ball-polyhedron in E3 is generated by at least three balls, then it possesses vertices, edges and faces. Finally, a ball-polyhedron is called a standard ball-polyhedron if its vertices, edges and faces (together with the empty set and the ball- polyhedron itself) form an algebraic lattice with respect to containment. We note that not every ball-polyhedron of E3 is a standard one a fact, that is somewhat surprising and is responsible for some of the difficulties arising at studying ball-polyhedra in general (for more details see [bn] as well as [blnp]). To each of a ball-polyhedron in E3 we can assign an inner dihedral angle. Namely, take any point p in the relative interior of the edge and take the two balls that contain the two faces of the ball-polyhedron meeting along that edge. Now, the inner dihedral angle along this edge is the angle of the two half-spaces supporting the two balls at p. The angle in question is obviously independent of the choice of p. Finally, at each vertex of a face of a ball-polyhedron there is a face angle formed by the two edges meeting at the given vertex (which is in fact, the angle between the two tangent half-lines of the two edges meeting at the given vertex). We say that the standard ball-polyhedron P in E3 is globally rigid with respect to its face angles (resp. its inner dihedral angles) if the following holds: If Q is another standard ball-polyhedron in E3 whose face-lattice is isomorphic to that of P and whose face angles (resp. inner dihedral angles) are equal to the corresponding face angles (resp. inner dihedral angles) of P , then Q is congruent to P . A ball-polyhedron of E3 is called triangulated if all its faces are bounded by three edges. It is easy to see that any triangulated ball-polyhedron is in fact, a standard one. The following statement has been proved in [bn]. Theorem. Let P be a triangulated ball-polyhedron in E3. Then P is globally rigid with respect to its face angles.

The related question below is still open. 4

Problem. Let P be a triangulated ball-polyhedron in E3. Prove or disprove that P is globally rigid with respect to its dihedral angles. It would be natural and interesting to investigate the same problems in spherical as well as in hyperbolic 3−space.

Bibliography [blnp] K. Bezdek, Zs. L´angi, M. Nasz´odi and P. Papez, Ball-polyhedra, Discrete Comput. Geom. 38 (2007), 201–230. [bn] K. Bezdek and M. Nasz´odi, Rigidity of ball-polyhedra in Euclidean 3-space, European J. Combin. 27 (2006), 255–268. A.3 Bobenko, Alexander Here I select several problems in rigidity I am interested in: 1. Which polyhedral surfaces composed of planar polygons (in particular of planar quadrilaterals) admit finite deformations? 2. Delaunay unfolding. Can the boundary of a convex polytop be unfolded into the plane without self-overlap by cutting the surface along adges of the Delaunay tesselation T of the boundary? (Note that the Delaunay tesselation is determined by the metric). In particular if all the faces of a polytop are acute triangles can it be unfolded by cutting alonfg the edges of the ? 3. Koenigs nets. are nets of planar quadrilateral with the combinatorics of the square grid such that the intersection points of the diagonals of four neighboring quads are coplanar. They possess remarkable transformations and rigidity properties to be studied. A.4 Dolbilin, Nikolay Here I select among them the most interesting topics. B. Point 1.. Given a convex polyhedron, whether there is at least one its “edge” unfolding homeomorphic to a disc? This problem has become of especial interest because of a quite new result. In 2006 I suggested to prove (or disprove) the so-called Anti-Durer¨ conjecture: Let P be a convex polyhedron and c(P ) minimal number of non-overlapping components in edge unfoldings of P , then C = sup c(P ) over all convex polyhedra is ∞. The well-known conjecture (”the Durer¨ conjecture”) says that in a set of convex polyhedra C = sup c(P ) = 1. I, in person, cannot say what answer is more probable: C equals either 1 or ∞. But the C seems to me not to be able to be in between. I offered to Alexei Tarasov and Alexei Glazyrin to prove the analogue the Anti-Durer¨ conjecture in the set of concave polyhedra. They proved it recently. B. Point 4. The problem of finding the maximum of the volume bounded by a closed surface, which is isometric (or submetric) to a given polyhedral sphere, could be called the isometric problem. In contrast to the classical isoperimetric problem, in the isometric problem the maximal volume is being searched for in a ”subspace” of all pairwise isometric surfaces of the whole space of all closed surfaces with same area. If the organizers are going to run a series of talks by participants, I could offer to the organizers to give one or two talks: 1. The new results on the Minkowski theorem on convex polyhedra and its applications. 2. Rigidity of zonohedra. 5

The last talk is going to be a survey of results on rigidity of surfaces with center symmet- rical faces obtained by M.Stan’ko, M.Stogrin, and N.Dolbilin. A typical result here sounds as follows: assume a polyhedral sphere with centrosymmetrical faces, which is immersed into space, is rigid (non-convexity and self-intersection under the immersion are allowed). A.5 Fernandez, Silvia Here is an open problem that may be of interest to some of the participants of the workshop. Let K be a strictly convex domain (i.e., a bounded subset of the plane such that if x and y are boundary points of K then the open segment xy is contained in the interior of K) with smooth boundary B(K). Consider a unit rigid rod (segment) R with endpoints p and q. Let R travel counter-clockwise along K in such a way that p and q are on the boundary of K at all times. More precisely, R rotates continuously and counter-clockwise 360 degrees, in such a way that p and q are always points of B(K). Of course, this is impossible sometimes. For example, if K is too small then R may not even fit inside K. And even if R fits inside K, it can suddenly get stuck. (This happens when the width of K in a certain direction is less than 1.) Now, assuming that R can complete a full turn around K, follow the trajectory of its end points along B(K). When K is “round” and “big” enough p and q simply move counter-clockwise along B(K). But interestingly enough, for some sets K, the points p and q may change directions while moving along B(K). My questions are: 1. Under which conditions on K, the points p and q change direction along the bound- ary of K a finite number of times? 2. Is it true that for every K the points p and q change direction along the boundary of K a finite number of times? 3. Is it possible to find, for every integer n > 0 a set K such that p changes direction at least n times while moving along the boundary of K? A.6 Fillastre, Francois A famous theorem of A.D. Alexandrov says that each metric of curvature K on the sphere with conical singularities of positive curvature can be realized as the induced metric on the boundary of a unique of the Riemannian space-form of curvature K. A similar result holds for metrics with negative singular curvatures, which are realized in a Lorentzian space-form (Rivin-Hodgson, 1993). My goal is to extend these resuts to the convex polyhedral realizations of metrics on surface of higher genus. There exists 10 different cases, classified by the genus, the constant sectionnal curvature and the sign of the singular curvatures. Ivan Izmestiev and myself are working on the last remaining case, that would lead to a unique general statement. To prove such results the main point is usually to prove (infinitesimal) rigidity results in different space-forms. The study can be bring from one space to another with the help of the so-called (infinitesimal) Pogorelov maps. I am interesting in learning about any method concerning (infinitesimal) rigidity of polyhedral/convex objects. A.7 Graver, Jack I have spent a great deal of time over the last 20 years attempting to find a combi- natorial characterization of 3-dimensiional generic rigidity. More recently my research has 6 been directed toward understanding the combinatorial structures of fullerenes - large carbon molecules. My main interest in attending this workshop is to catch up on the latest research in rigidity and to find new interesting problems to investigate. A.8 Izmestiev, Ivan I would like to learn from Jean-Marc Schlenker and Francois Fillastre about the Pogorelov map (both the infinitesimal and the finite version of it). A.9 Martin, Jeremy I’m interested in using combinatorial techniques to study the geometric invariants that come from rigidity theory. I typically think of these invariants as “local” constraints on the pieces of a geometric object (e.g., the bars in a bar-joint framework, or the facets of a polytope) that somehow come from the object’s “global” rigidity properties. For example, in previous work, I’ve looked at the constraints on the slopes of the edges in a plane embedding of a graph (by viewing them as the equations defining the space of all possible embeddings). It turns out that one can say quite a lot about those constraints using algebraic and combina- torial tools, such as Gr¨obner bases, Stanley-Reisner theory, and the Tutte polynomial. One of the goals of my joint project with Mike Develin and Vic Reiner was to broaden the scope of this kind of work from graphs to matroids; for any representable matroid, we constructed a space whose points are supposed to parametrize “photographs” of the matroid. I’d like to figure out how the defining equations of this space are given by the combinatorial structure of the matroid, and conversely how the combinatorics encodes actual geometry; for instance, the photo space construction (sort of) captures the fact that four nonzero vectors in a plane have a cross-ratio. I don’t have any other specific problems in mind (yet), but I’m very much looking forward to hearing other mathematicians’ ideas at the workshop. A.10 Nevo, Eran Stresses and the Colin de Verdi`ere’s graph parameter. Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. The conjecture below is a generalization of this result. Let µ(G) denote the Colin de Verdi`ere’s parameter of a graph G. Let G be a graph and let k be a positive integer. If µ(G) ≤ k then G is generically k-stress free. For k = 1, 2, 3, 4 this conjecture is true: Colin de Verdi`ere showed that the family {G : µ(G) ≤ k} is closed under taking minors for every k. Note that for the on r vertices µ(Kr) = r − 1. Now apply the following theorem: For 2 ≤ r ≤ 6, every Kr-minor free graph is generically (r − 2)-stress free. While Theorem fails for r ≥ 8, as is demonstrated for r = 8 by K2,2,2,2,2 and for r > 8 by repeatedly coning over it (see e.g. Song), Conjecture may hold. This conjecture implies

k+1 µ(G) ≤ k ⇒ e ≤ kv − ( 2 )

(where e and v are the numbers of edges and vertices in G, respectively) which is not known either. Further details can be found in my paper “On embeddability and stresses of graphs”, to appear in Combinatorica 27(4); an extended version on arXiv:math/0411009v1. 7 A.11 Panina, Gayane I’m very much interested in the interplay of the combinatorial rigidity (especially, theory of pseudo-triangulations) and the theory of hyperbolic virtual . A (far from complete) list of problems that arose in this respect (in Russian) is available at http://fizmatclub.spb.ru/courses/virtualpolyhedraandpseudotriangulations. Here are three of them. A. A pointed bar-and-joint mechanism in the plane produces an expansive motion of its vertex set. This fact has been used in the construction of an analogue of secondary polytope for pointed tilings. What are parallel statement and consequences for pointed bar-and-joint mechanism in the sphere? (The mechanism is supposed not to fit a hemisphere.) B. What graphs possess a pointed embedding in the sphere S2? At least, describe all combinatorial types of pointed triangulations of S 2. (They do exist, there is no misprint.) C. The group of rational virtual polytopes is known to be canonically isomorphic to the limit of Picard groups of toric varieties. The isomorphism maps to very ample vector bundles. What is an algebraic characterization of hyperbolic virtual polytopes in this frame- work? A.12 Rote, Gunter A) I am interested in sortest paths on the surface of convex polytopes in more than three dimensions. Miller and Pak have developed an algorithm that computes the “sortest path map” from a given point in time polynomial in the complexity of this map, but it is unknown whether this complexity is polynomial in the input, in, say, d=4 dimensions. Combinatorially, this amounts to asking, how many combinatorially different shortest paths can emanate from a given point? A path is combinatorially characterized by the sequence of ridges ((d-2)-dimensional faces) that it traverses. B) Another (notorious open) question that I like is whether every 3-polytope has a non- overlapping edge unfolding. I have thought about the infinitesimal limit of such unfoldings, when one considers a path of the surface, and by a linear transformation, makes it flatter and flatter. A.13 Sabitov, Idzhad There are three problems which have interested me for a long time. 1) Fixation of a polyhedron. When we want to find a simplicial polyhedron (that is one having only triangular faces) with a given combinatorial structure and given length of edges we compose a system of equations 2 2 2 2 (xj − xi) + (yj − yi) + (zj − zi) = lij, (1) expressing the square of length of edges with the vertices numbered by some subscript indices. However to any solution P = (x1, . . . , zn) of such a system (where n is the number of vertices) there corresponds a whole family of solutions obtained by a motion of P in space, and if we have another solution P1 of (1) we don’t know without a special study whether this solution 8 is a congruent copy of P or it is a new polyhedron isometric to P and noncongruent to it. To avoid these “parasitic” solutions we need to introduce some additional conditions forbidding to move, at least continuously, any obtained solution. In other words, we would like to replace the operation expressed abstractly as “factorization relative to the group of motions” by some equations. For example, the equations n n n X xi = 0, X yi = 0, X zi = 0 i=1 i=1 i=1 help to avoid the parallel translations. The question is to find such equations permitting to exclude rotations and valid for any polyhedra whose vertices are not on a straight line (for more of details see my article in Sbornik:Mathematics 189:10 (1998), p.1533-1561). It is known how to do this for polygons on the plane. Such a question arises any time we want to reconstruct a finite set of points starting from the distances between points. 2) We call canonical a polynomial equation for volume of a polyhedron having a minimal degree (f.e. the canonical polynomial equation for the volume of a has degree 16). In the case of polyhedron homeomorphic to the sphere there is a theorem affirming the existence and unicity of such an equation with an algorithm to find it (see Astrelin and Sabitov, Russian Math. Survey, 54(2) (1999), 430-431). The question is to find a similar (or other?) theorems for polyhedra of genus ≥ 1. 3) To prove the generic infinitesimal rigidity of polyhedra in any n-space or, at least, their generic infinitesimal rigidity of second order (formally even in 3-space the generic infinitesimal rigidity is proved only for polyhedra of genus g = 0 and g = 1). If we have such a proof it would give us a new simple proof for the affirmation on generic rigidity of polyhedra in any n-space (the proof by Fogelsanger (1988) of this affirmation is very heavy to understand and to verify). A.14 Schlenker, Jean-Marc Rigidity of weakly convex polyhedra Convex polyhedra in Euclidean 3-space are known to be rigid since Cauchy. They are also infinitesimally rigid, as was first proved by Dehn. We are interested in the question whether the convexity hypothesis can be weakened to the hypothesis that the vertices are in convex position. It is then necessary to suppose also that the polyhedron is decomposable (can be cut into convex pieces without adding an interior vertex). In a recent work in collaboration with Bob Connelly we proved that this conjecture holds in two special cases, and in particular for suspensions which contain their north- south axis. More recently we proved, with Ivan Izmestiev, that the conjecture also holds for polyhedra which are “codecomposable” (apparently a very weak hypothesis, satisfied in almost all known examples). This last result is related to the behavior of the Hilbert-Einstein (aka Regge) function on triangulated polyhedra. We prove that the Hessian of this function, for deformations fixing the boundary metric, has a simple behavior. A.15 Schulze, Bernd I am interested in rigidity properties of frameworks that possess non-trivial symmetries. Consider, for example, a symmetry group C3 generated by a 3-fold rotation C3. Given a 9 graph G = (V, E) and a homomorphism Φ : C3 → Aut(G), we conjecture that a 2-dimensional realization (G, p) which is (C3, Φ)-generic (i.e., ‘generic’ within the set of all realizations of G that have symmetry C3 and Φ) is a rigidity circuit if and only if |E| = 2|V | − 2, |E(U)| ≤ 2|U| − 3 for all U ⊆ V with |U| ≥ 2 and exactly one vertex of G is fixed by Φ(C3) if and only if G can be constructed from K4 by a sequence of symmetric vertex splits and symmetric gluing. Moreover, we conjecture that if G is 3-connected, satisfies the counts |E| = 2|V | − 2 and |E(U)| ≤ 2|U| − 3 for all U ⊆ V with |U| ≥ 2 and there exists exactly one vertex of G that is fixed by Φ(C3), then all (C3, Φ)-generic realizations of G are globally rigid in the plane. A.16 Servatius, Brigitte Autopolarity and self-duality: Self-dual polyhedra were studied by several authors. The one remaining open question is which self-dual polyhedra have an auto-polar embedding and how to find it. Polyhedral allostery: Find at least interesting examples of graphs, formed by gluing polyhedra together, so that in any 3-d realization a small change in length of one edge forces a change in length of a “far away” edge. Toward the molecular conjecture: Find a class of graphs G so that G2 is rigid in 3-d. A.17 Stachel, Hellmuth I’m particularly interested on higher order infinitesimal flexibility of linkages and poly- hedra as well as on the kinematics of flexible polyhedra and overconstrained linkages in Euclidean and non-Euclidean spaces of dimension n ≥ 2. A.18 Tabachnikov, Sergei I would suggest to discuss the following problems related to my recent work and general research interests: 1. Affinely and projectively self-dual polygons and polyhedra. Surprisingly, not much is known about polygons in the projective plane which are projectively equivalent to their projective duals (for example, every pentagon is such). In a recent joint paper with D. Fuchs we gave a description of such polygons. It is interesting to extend this to polygons and to polyhedra in multi-dimensional projective and affine spaces. 2. Geometry of bi-equilateral (or bicycle) polygons. A polygon is bi-equilateral if its sides have equal lengths and its k-diagonals also have equal length. A continuous version of such polygons, are the boundaries of two dimensional bodies that float in equilibrium in all position. These curves also correspond the case when it is impossible to determine the direction of motion from the tracks of the rear and the front wheels of the bicycle. It is interesting to study such polygons in the spherical and hyperbolic metric, in multi- dimensional setting, and also to consider their polyhedral analogs. 3. Sub-Riemanian geometry of polygons and billiards. The space of polygons carries a completely non-integrable distribution defined by the optical condition “angle of incidence equals angle of reflection”: nth vertex Vn is allowed to move along the bisector of the exterior angle made by the sides Vn−1Vn and VnVn+1. An analog of this distribution exists for outer billiard system as well. The recent study of this distribution made it possible to construct 10 inner and outer billiards with invariant curves consisting of periodic points (analogs of figures of constant width) and to contribute to the famous open problem: the set of periodic billiard trajectories has zero measure. Open problems abound. For example, are there plane convex billiards, other than a circle or an ellipse, for which invariant curves of different periods (say, 2 and 3) can coexist?

4. Origami hyperbolic paraboloid. A well known origami construction produces a surface that looks like a hyperbolic paraboloid. Is it really and why? What happens if the parameters of the construction vary (say, 4-fold symmetry is destroyed or replaced by another kind of symmetry)? A.19 Tarasov, Alexey I am interesting in new ideas which can possibly help to solve Duhrer conjecture. Also I will be glad to explain my ideas including counter-example to Schlenker’s question, whether the conjecture extends to general convex metrics with convex faces. A.20 Thurston, Dylan Steven Gortler, Alex Healy and I recently completed a characterization of which graphs are generically globally rigid in d-dimensional Euclidean space, proving a conjecture of Con- nelly. Although the characterization is explicit and checkable, it is not very combinatorial in nature, and it is difficult to tell from first examining a graph whether it satisfies the condition. It would be interesting to find more interesting examples of graphs which are or are not generically globally rigidy, beyond Connelly’s example of K5,5 in 3 dimensions, and to try to find more combinatorial charcterizations. Our preprint “Characterizing Generic Global Rigidity” is available on the arXiv: 0710.0926 There are a number of other questions which have been relatively little considered. For instance, suppose we have a graph which is generically globally rigid in d dimensions. Which of the generic configurations in d dimensions remain globally rigid when considered as a configuration in d + 1 dimensions? Unlike the previous question of generic rigidity, this does depend on which generic configuration we consider. A.21 Whiteley, Walter There are a number of problems which seem amenable to work in the workshop. Some connect directly to topics raised by other participants. Techniques for Generating Globally Rigid Frameworks Let me list a couple of conjectures from a larger collection of conjectures we have pulled together. If G is generically globally rigid in IRd, and G0 is created from G by a vertex split with both new vertices of degree at least d + 1, then G is generically globally rigid in IRd. This has been verified in the plane, by the work of Jordan and Jackson. If G is generically globally rigid in IRd, and G0 is created from G by coning with a new vertex, the G0 is generically globally rigid in IRd+1. This is equivalent to the conjecture that if G is globally rigid in IRd then G is globally rigid in Sd. This in turn should connect to the Pogerlov map. We note that First-order Rigidity is projectively invariant, and does not the depend on the choice of Cayley-Klein metric (Hyperbolic, Spherical, deSitter, ... ) imposed at any 11 configuration. So these conjectures do match the simple necessary conditions of redundant rigidity and connectivity. Determination of facial angles of polyhedra by dihedral angles A theorem of Minkowski shows that, given the areas and normals of the faces of a spherical polyhedron, with one basic equality, give a unique convex polyhedron. There are also results of Stokker and Karcher [karcher, stokker] about face angles of polyhedra giving relative uniqueness of the convex polyhedron (up to similarity). From the modern analysis of the reciprocal diagrams of Maxwell (and Cremona) [maxwell], we have some additional tools, and insights, into the isomorphism of self-stresses on the Gaussian sphere, of normals (as vertices) and fixed dihedral angles (as length con- straints) - the spherical diagram - and realizations as plane faces polyhedra (up to translation and scaling). In particular, for convex polyhedra (where the spherical diagram is convex: planar and has one self-stress with all entries the same sign), the dimension of the space of self-stresses corresponds to the space of Minkowski sums / factors of the polyhedron. A key conjecture of Stokker is that, for convex polyhedra, the dihedral angles gives the facial angles. This is equivalent to asserting that, for convex polyhedra the spherical diagram is unique within the class of convex spherical diagrams. There are indications that this may not be true. The proposed task is to find a counter-example, using techniques of rigidity. Response to small changes in lengths If we have an isostatic framework G(p) (e.g. a triangulated spherical polyhedron), and we treat the edges as springs with that rest length, there is a key question of how it responds to distortion. That is, if we distort the configuration to q, and assume the bars exert forces proportional to the change in length (or the change in length squared), will the structure return to exactly the initial configuration? Put so generally, this cannot be true as the framework is not globally rigid (by the work of Hendrickson [hendrickson]) and the other realizations with the same edge lengths will also be in equilibrium. However, if the distorted configuration is also convex, it cannot be in equilibrium, by the uniqueness and independence theorems of Cauchy and Dehn. Will the framework be restored to the original (convex) position (up to translation, rotation, reflection), remaining at all times convex? If this is not true in full generality, how small does the distortion have to be to ensure that it is true? This question is related to more general questions about control of formations of au- tonomous agents through the use of isostatic frameworks as link-lengths [formations]. It may also connect to the semidefinite programming work described by Yinu Ye. Polarity on planes, points, surfaces .... Recall the rigidity and polarity theorem of [polarityI] which essentially shows: Given a spherical polyhedron P then there is a projectively polar polyhedron P ∗, with dual vertices, faces etc. Then if the original polyhedron and the polar polyhedron have their faces triangulated (in the sense of Alexandrov’s Theorem [infpI]), so that each plane framework is isostatic in its plane, then the first polyhedral framework is isostatic if and only if the second polyhedral framework is isostatic, even if both are non-convex. Our inspection of results for smooth surfaces, with exceptional pointed vertices, or exceptional planar regions, indicates that the known rigidity results come in polar pairs 12

(with the exceptional pointed vertices being polar to the exceptional planar regions). We conjecture that there is a general polarity theorem which will simplify the work on smooth surfaces both for static rigidity and infinitesimal rigidity (which no longer coincide). Bibliography [maxwell] Henry Crapo and Walter Whiteley Spaces of stresses, projections and paral lel drawings for spherical polyhedra, Contributions to Algebra and Geometry 35 (1994), 259281. [formations] T. Eren, W. Whiteley, B. Anderson, A. Morse, and P. Belhumeur: Operations on rigid formations of autonomous agents. Communications in Information and Systems 3 (2004), 223-258. [hendrickson] Bruce Hendrickson: Conditions for unique graph realizations SIAM J. Comput (1992) 21, 65-84 [stokker]J.J. Stokker: Geometric problems concerning polyhedra in the large; Com. Pure and Applied Math. (1968) 21, 119-168. [karcher]H. Karcher: Remarks on polyhedra with given dihedral angles; Com. Pure and Applied Math. (1968) 21, 169-174. [polarityI] W. Whiteley: Rigidity and Polarity I: Statics of sheet structures, Geometrie Dedicata 22 (1988), 229-302. [infpI] W. Whiteley: Infinitesimally rigid polyhdra I: statics of frameworks, Trans. A.M.S., 285 (1984), 431-465. A.22 Ye, Yinyu We have worked on semidefinite programming (SDP) based approaches for the position estimation problem in Euclidean distance geometry such as graph realization and sensor net- work localization. We develop an SDP model and use the duality theory to derive necessary and/or sufficient conditions for whether a network is “uniquely realizable or localizable” or not, when the incomplete distance measures are accurate. We also present error analyses of the SDP solution when the distance measures are noisy. Furthermore, we develop an algorithm such that large-scale networks can be realized efficiently, and demonstrate com- putational results to show the effectiveness of the SDP relaxation model.