PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA

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1 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA NICHOLAS BARVINOK AND MOHAMMAD GHOMI Abstract. A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. Thus Dürer s conjecture does not hold for pseudoedge unfoldings. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability criterion for almost flat convex caps due to Tarasov. 1. Introduction A convex polyhedron is the boundary of the convex hull of finitely many points in Euclidean space R 3 which do not all lie in a plane. A well-known conjecture [6], attributed to the Renaissance painter Albrecht Dürer [7], states that every convex polyhedron K is unfoldable, i.e., it may be cut along some spanning tree of its edges and isometrically embedded into the plane R 2. Here we study a generalization of this problem to pseudo-edges of K, i.e., distance minimizing geodesic segments in K connecting pairs of its vertices. A pseudo-edge graph E of K is a 3-connected embedded graph composed of pseudo-edges of K, with the same vertices as those of K, and with faces which are convex in K, i.e., the interior angles of each face of E are less than π. Cutting K along any spanning tree T of E yields a simply connected compact surface K T which admits an isometric immersion or unfolding u T : K T R 2. If u T is one-to-one for some T, then we say that K is unfoldable with respect to E. Our main result is as follows: Theorem 1.1. There exists a convex polyhedron K with a pseudo-edge graph with respect to which K is not unfoldable. Thus one may say that Dürer s conjecture does not hold in a purely intrinsic sense, since it is not possible to distinguish a pseudo-edge from an actual edge by means of local measurements within K. On the other hand, by Alexandrov s isometric embedding theorem [1], any convex polyhedron is determined up to a rigid motion by its intrinsic metric. So edges do indeed exist intrinsically, although Alexandrov s proof is not constructive and does not specify their location. A more constructive Date: January 16, 2018 (Last Typeset) Mathematics Subject Classification. Primary: 52B10, 53C45; Secondary: 57N35, 05C10. Key words and phrases. Edge unfolding, Dürer conjecture, almost flat convex cap, prescribed curvature, weighted spanning forest, pseudo-edge graph, isometric embedding. Research of the second named author was supported in part by NSF Grant DMS

2 2 NICHOLAS BARVINOK AND MOHAMMAD GHOMI approach has been studied by Bobenko and Izmestiev [4] but that too does not predict the position of the edges. In short, the edges of convex polyhedra are not well understood from the point of view of isometric embeddings, and, in light of the above theorem, it would now be even more remarkable if the conjecture holds. Theorem 1.1 was first announced in 2008 in a highly original and hitherto unpublished manuscript by Alexey Tarasov [20]. Unfortunately, typographical and computational errors, and other expository issues, make it difficult to understand or verify all the claims in that work. Nevertheless we can confirm that certain key ideas in [20] were correct, and employ these (Section 4) in the proof of the above theorem. The example we construct, however, is far more simple as it contains only 176 as opposed to over 19,000 vertices. The construction in [20] is quite convoluted and we have not been able to check it thoroughly. The polyhedron K in Theorem 1.1 is obtained by arranging 4 congruent almost flat convex caps over the faces of a regular tetrahedron. These caps have 43 interior vertices each with prescribed curvature and projection. They are constructed via a result of Pogorelov on convex caps with prescribed curvature as we describe in Section 2. In Section 3 we study the pseudo-edges induced on a convex cap C by the edge graph G of convex subdivisions of the polygon at the base of C. Then in Section 4 we describe a necessary condition, due to Tarasov, for unfoldability of C in terms of spanning forests of G. Next in Section 5 we construct a convex subdivision of an equilateral triangle which does not satisfy Tarasov s criterion. Consequently, sufficiently flat convex caps constructed over this subdivision fail to be unfoldable with respect to the induced pseudo-edge graph. In Section 6 we assemble 4 such caps to construct K. The edge unfolding problem for convex polyhedra was first explicitly formulated by Shephard [19] in 1975, and since then has been advertised in several sources, e.g., [5, 6, 10, 13, 17, 21]. The conjecture that the answer is yes, i.e., all convex polyhedra are unfoldable, appears to be first stated by Grünbaum [11] in The earliest known examples of unfoldings of convex polyhedra were drawn by Dürer [7] in 1525, all of which were nonoverlapping. Hence the unfolding problem or conjecture are often associated with his name. For more background, references, and a positive recent result see [9] where it is shown that every convex polyhedron becomes unfoldable after an affine transformation. See also O Rourke [14, 16] for other recent positive results concerning unfoldability of certain convex caps. As far as we know, Theorem 1.1 is the first hard evidence against Dürer s conjecture. 2. Convex Caps with Prescribed Boundary and Curvature A (polyhedral) convex cap C R 3 is a topological disk which lies on a convex polyhedron and whose boundary C lies in a plane H, while its interior C \ C is disjoint from H. The normal cone N p (C) of C at an interior point p is the convex cone generated by all outward normal vectors to support planes of C at p. The unit normal cone N p (C) is the collection of unit vectors in N p (C). The curvature of C at p is defined as k(p) = k C (p) := 2π σ(n p (C)),

3 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 3 where σ denotes the area measure in the unit sphere S 2. Let π : R 3 R 2 denote the projection into the first two coordinates. A set X R 3 is a graph over R 2 provided that π is one-to-one on X, and X R 2 [0, ). A convex polygon P is the convex hull of finitely many points in R 2 which do not all lie on a line. We say that a convex cap C is over P provided that C is a graph over R 2 and C = P. We need the following result of Pogorelov [18, Lem. 1, p. 65], see also Pak s lecture notes [17, Thm 35.7]. Yet another proof of the following result, based on Alexandrov s mapping lemma, may be found in an earlier draft of this work [2, Thm 2.1]. Lemma 2.1 (Pogorelov [18]). Let P be a convex polygon, p i, i = 1,..., n, be points in the interior of P, and β i > 0 with i β i < 2π. Then there exists a unique convex cap C over P with interior vertices v i such that π(v i ) = p i, and k(v i ) = β i. A convex subdivision of a convex polygon P is a subdivision of P into convex polygons each of whose vertices either lies in the interior of P or coincides with a vertex of P (we assume that the interior angles of P at all its vertices are less than π). If G is the (edge) graph of a convex subdivision of P, then by an interior vertex p i of G we mean a vertex of G which lies in the interior of P. We assume that the angles of incident edges of G at p i are all less than π. We say that G is weighted if to each of its interior vertices there is associated a number α i > 0 with i α i = 1. Let the total curvature k(c) of a convex cap C be the sum of the curvatures of its interior vertices. Lemma 2.1 immediately yields: Corollary 2.2. Let P be a convex polygon, and G be the weighted graph of a convex subdivision of P, with interior vertices p i and coefficients α i. Then for any 0 < β < 2π there exists a convex cap C β over P with interior vertices v i such that π(v i ) = p i and k(v i ) = β i := α i β. In particular k(c β ) = β. 3. Pseudo-Edge Unfoldings of Almost Flat Convex Caps In this section we fix P, G, and α i to be as in Corollary 2.2, and aim to study the corresponding convex caps C β for small β. In particular we will show that G gives rise to a unique pseudo-edge graph G of C β (Proposition 3.4) and study the corresponding unfoldings of C β in relation to P (Proposition 3.5) The induced pseudo-edge graph of C β. First we check that as β 0, C β P. More precisely, if d β denotes the intrinsic distance in C β, then we have: Lemma 3.1. As β 0, d β (x, y) π(x) π(y), for all x, y C β. Proof. As β 0, the maximum height of C β goes to zero. If not, there exists a sequence β k 0 such that the maximum height of C k := C βk is bounded below by h > 0. So, for some i, the height of the vertex vi k of C k which projects onto p i is bounded below by h. Let o be the point of height h above p i, and C be the convex cap formed by line segments connecting o to points of P. Since C lies below C k, every support plane of C at an interior vertex is parallel to a support plane of C k at an interior vertex. So k(c k ) k(c ) > 0, which is the desired contradiction since k(c k ) = β k 0.

4 4 NICHOLAS BARVINOK AND MOHAMMAD GHOMI Now let L be the line segment connecting π(x), π(y), and L be the corresponding curve in C β connecting x, y, such that π(l) = L. Then d β (x, y) length(l). But L is the graph of a convex function over L which converges to 0. Thus length(l) length(l) = π(x) π(y). So the limit of d β (x, y) is not bigger than π(x) π(y). On the other hand d β (x, y) x y π(x) π(y). So d β (x, y) π(x) π(y). A polyhedral disk D is a topological disk composed of a finite number of convex polygons identified along their edges. We say that D is locally flat if the total angle at its interior vertices is 2π. An isometric immersion f : D R 2 is a locally oneto-one continuous map which preserves distances between points on each face of D. If f is one-to-one everywhere, then we say that it is an isometric embedding. Lemma 3.2. Let D be a locally flat polyhedral disk. Suppose that the total angle at each of the boundary vertices of D is less than 2π. Then there exists an isometric immersion D R 2. Proof. The angle condition along D ensures that each point of D has a neighborhood which may be isometrically embedded into R 2. Since D is simply connected, a family of these local maps may be joined to produce the desired global map, e.g., see the proof of [8, Lem. 2.2] for further details. By a geodesic in C β we mean the image of a continuous map γ : [a, b] C β such that length[γ] = d β (γ(a), γ(b)). For any set X R 2, and r > 0, U r (X) denotes the set of points in R 2 which are within a distance r of X. Further set δ := min i j p i p j where p i, p j range over all vertices of G. Lemma 3.3. If β is sufficiently small, then to each edge e of G there corresponds a unique geodesic e of C β whose end points project into the endpoints of e, and π(e) U δ (e). Proof. Let x, y C β be points which project into end points of e and Γ be a geodesic in C β connecting x and y. By Lemma 3.1, length(γ) π(x) π(y). Thus eventually π(γ) U δ (e). Suppose that there exists another geodesic Γ in C β connecting x and y. Then again we have π(γ ) U δ (e) for β small. Let V C β be the region with π(v ) = U δ (e). Then Γ, Γ V. Now if Γ Γ, then there exists a simply connected domain D V bounded by a pair of segments Γ 0 and Γ 0 of Γ and Γ respectively. Then D admits an isometric immersion f : D R 2 by Lemma 3.2. But f maps Γ 0 and Γ 0 to straight line segments with the same end points. Hence f(γ 0 ) = f(γ 0 ). In particular f is not locally injective at the points of D where Γ 0 and Γ 0 meet, which is a contradiction. A convex polygon X in C β is a region bounded by a simple closed curve composed of a finite number of geodesics meeting at angles which are less than π with respect to the interior of X. A convex subdivision of C β is a subdivision into convex polygons whose interiors contain no vertices of C β, and whose vertices are vertices of C β. A pseudo-edge graph of C β is the edge graph of a convex subdivision.

5 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 5 Proposition 3.4. For β sufficiently small, there exists a unique pseudo-edge graph G of C β such that π(g) U δ (G). Furthermore, there exists a canonical homeomorphism f : P C β such that f is identity on P, f(g) = G, and f converges to the identity map on P as β 0. Proof. Existence and uniqueness of G follow quickly from Lemmas 3.3 and 3.1. To construct f, let G T be the triangulation of G given by connecting the center of mass of each nontriangular face of G to its vertices. By Lemma 3.3, there exists a unique triangulation G T of G such that π(g T ) U δ (G T ) and the vertices of G T project onto the vertices of G T. For any vertex p of G T let f(p) := π 1 (p) C β. For any triangle of G T, let be the triangle of G T whose vertices project onto the vertices of. Let u : R 2 be the isometric embedding of into a planar triangle, given by Lemma 3.2. We define a mapping g : as follows. For any x, let (x 1, x 2, x 3 ) be the barycentric coordinates of x with respect to the vertices v 1, v 2, v 3 of. Let g (x) be the point of whose barycentric coordinates with respect to the vertices v i := u π 1 (v i ) of are (x 1, x 2, x 3 ). Finally set f(x) := u 1 (g (x)), where is a triangle of G T which contains x Cut forests and unfoldings of C β. A tree is a connected graph without cycles. A subgraph F of G is called a cut forest if (i) F is a collection of disjoint trees which contain all the vertices of G in the interior of P, and (ii) each tree of F contains exactly one vertex of P ; see the middle diagram in Figure 1. By Lemma G F P C β,f Figure , to each cut forest F of G there corresponds a unique cut forest F of G, assuming β is small. Let C β,f be the surface obtained from C β by cutting it along F, i.e., take the disjoint collection of the faces of G and glue them together pairwise along all their common edges which do not belong to F. By Lemma 3.2 there exists an isometric immersion, or unfolding map u: C β,f R 2. We assume that u fixes a designated edge e 0 of C, and locally maps C β,f to the same side of e 0 where P lies. Let c: C β,f C β be the natural covering map which sends each face of C β,f to the corresponding face of C β, and f : P C β be the homeomorphism given by Proposition 3.4. Then (1) φ := u c 1 f,

6 6 NICHOLAS BARVINOK AND MOHAMMAD GHOMI is a multivalued mapping P C β,f := u(c β,f ). which converges to the identity on P as β 0. For any point x P, we set x := φ(x). Note that φ is single valued on P \ F and is one-to-one on the interior of each face Φ of G. Let Φ indicate the corresponding face of C β,f, i.e., the closure of φ(int(φ)). Note that φ yields a bijection φ Φ between Φ and Φ, by setting φ Φ := φ on int(φ) and extending the map continuously to the boundary of Φ. For any x Φ, let x Φ := φ Φ (x) denote the corresponding point of Φ. As β 0, x Φ x for all faces Φ of G which contain x. Furthermore φ Φ converges uniformly to the identity map on Φ, because it depends continuously on β. Thus it follows that: Proposition 3.5. For every ɛ > 0, there exists β > 0 such that for all x P dist(x, x ) ɛ; Furthermore, for any pair of points x, y which lie in the same face Φ of G (x y) (x Φ y Φ) ɛ x y. 4. Tarasov s Monotonicity Condition As in the last section, let P be a convex polygon, G be the graph of a fixed convex subdivision of P with weights α i, and C β be the corresponding convex cap over P given by Corollary 2.2. Here we describe Tarasov s criterion for unfoldability of C β with respect to the induced pseudo-edge graph G given by Proposition 3.4. A path Γ in G is a sequence of adjacent vertices. We say that Γ is simple if all of its vertices are distinct. If F is a cut forest of G, then each point p of G which lies in the interior of P may be joined to P with a unique simple path Γ p in F, which we call the ancestral path of p. This induces a partial ordering on points of G as follows: we write x y, for x, y G and say that x is an ancestor of y, or y is a descendant of x, if x Γ y. In particular note that x x. If x y and x y then we say y is a strict descendant of x, or x is a strict ancestor of y, and write x y. Further we adopt the following convention: for any x F, we write i x provided that p i x, where p i denote the vertices of G. For any x G, we define the center of rotation of x as the center of mass of its descendant vertices with respects to the weights α i : c x := α 1 x i x α i p i, where α x := i x Roughly speaking, c x is the limit, as β 0, of the pivot point about which x rotates to generate x defined in the last section (see Note 4.3). Every interior vertex p i of G has a unique adjacent vertex p i in F which is its parent or first strict ancestor which is a vertex. We also refer to p i as a child of p i. α i.

7 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 7 A cut forest F of G is called monotone (in the sense of Tarasov), if for every interior vertex p i of G we have (2) p i p i, p i c i 0, where c i := c pi. So, if p i c i and 0 p i p ic i π denotes the angle between the vectors p i p i and c i p i, then we have (3) p i p i c i π/2. In particular, every parent must be further away from the center of rotation of its child than the child is: (4) p i c i > p i c i. Hence the term monotone. Some other notions of monotonicity have been studied recently by O Rourke [14,16], and Lubiw and O Rourke [12] for cut forests of convex polyhedral disks; see also [9] for yet another monotonicity notion. Recall that C β,f is the image of the unfolding map u: C β,f R 2. We say that C β,f is simple, if and only if u is injective. If C β,f is simple for some cut forest F of G, we say that C β is unfoldable with respect to G. If G admits no monotone cut forests, then we say that G is non-monotone. The rest of this section is devoted to establishing the following result which parallels [20, Thm. 1]. Theorem 4.1. If G is non-monotone, then C β is not unfoldable with respect to G, for β sufficiently small. We prove the above theorem via the same general approach indicated in [20], although we correct a number of errors or ambiguities, and make many simplifications. Fix a cut forest F of G. If x F is not a vertex, x consists of precisely two elements: x R and x L defined as follows. Orient the edge e of F containing x from the child to the parent vertex. Then we can distinguish the faces Φ R, Φ L of G which lie to the right and left of e respectively. We set x R := x Φ R, and x L := x Φ L. Let J be the π/2-clockwise rotation about the origin of R 2, and set (5) c x := x + J(x R x L ), where β x := α x β = β i. β x i x The next observation parallels [20, Lem. 1]. Lemma 4.2. For every x F, c x c x, as β 0. Proof. Let F x := {y F y x}, and Γ be a polygonal Jordan curve in P which encloses F x and intersects F only at x; see the left diagram in Figure 2. Then Γ := φ(γ) is a polygonal path connecting x R and x L. Let U be the region bounded by Γ which contains F x, and S i U be simple polygonal paths which connect each p i x to x without intersecting each other and Γ; see the middle diagram in Figure 2. We are going to reindex S i and p i as follows. Let σ be a circle centered at x with sufficiently small radius so that it intersects Γ only twice, and each S i only once.

8 8 NICHOLAS BARVINOK AND MOHAMMAD GHOMI Orient σ counterclockwise, reindex S i, from i = 1,..., k, in order that they intersect σ U, and then reindex p i accordingly. x x L x x 2,L x 2,R x R Γ S 1 S 2 Γ Γ p 2 β 1 β 2 p 2 β 3 S 3 U U p 1 p 3 U p 1 p 3 Figure 2. Now let S := i S i be the resulting spanning tree for vertices of U. Then S := f(s) is a tree on U := f(u). Let U S denote the topological disk obtained by cutting U along S, c S : U S U be the corresponding covering map, u S : U S R 2 be an unfolding given by Lemma 3.2, and define the multivalued mapping θ : U R 2 by θ := u s c 1 S f. Comparing this definition with that of φ given by (1) shows that Γ := θ(γ) is congruent to Γ. Indeed Γ, Γ are determined, up to a translation, by the edge lengths of Γ and its interior angles with respect to U. So we may assume that Γ = Γ. Further note that θ converges to the identity map on U, just as φ does by Proposition 3.5. Thus, if we set x := θ(x), then x x, as β 0. Each p i in U has a single image p i under θ, while there are two images of x under θ corresponding to each S i, which are denoted by x i,l and x i,r ; see the right diagram in Figure 2. These may be defined similar to the way we defined x L and x R, by extending S to a triangulation of U. We claim that (6) c x = α 1 x i x α i p i, where p i := x + J(x i,r x i,l ) β i. Indeed, since Γ = Γ, and due to our reindexing of S i, we have x R = x 1,L, x L = x and x i,r = x i+1,l, for 1 i k 1. Thus, since β i = βα i = β x αx 1 α i, x R x L = i x (x i,r x i,l) = β i J(x p i ) = J(β x x β i p i ) = β x J(x c x ). i x i x Applying J to the far left and right sides of the last expression yields (6). Next note that, since x i,l p i x i,r = β i, elementary trigonometry yields that i,l ) p i = x i,m + J(x i,r x 2 tan(β i /2) see Figure 3. Thus we have i,r, where x i,m = x i,l + x 2 ; k,l,

9 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 9 x i,l p i β i S i,l x i,m S i,r x i,r Figure 3. p i p i = (x i,m p i )(1 2 tan(β i /2)/β i ) + (x x i,m) + (p i p i ). Since the right hand side vanishes, as β 0, it follows that p i p i, and consequently c x c x as desired. Note 4.3. Let R p,θ : R 2 R 2 denote the clockwise rotation about the point p by the angle θ. As we discussed in the proof of Lemma 4.2, x i,r = R p i,β i (x i,l ). So, since p i p i, as β 0, x R = R p 1,β 1 R p k,β k (x L) R p1,β 1 R pk,β k (x L) = R px,θ x (x L), for some p x R 2 and θ x [0, 2π). It is well-known that [15, Lem. 1], as β 0, θ k i=1 β i = i x β i = β x, and p x β 1 x k i=1 β i p i = α 1 x α i p i = c x. So c x is the limit of the cumulative pivot point of descendant vertices of x, which is the justification for the term center of rotation. Proof of Theorem 4.1. Let F be a cut forest of G. Then there exists an interior vertex p i of G which does not satisfy (2) and will be fixed henceforth. Let Φ L and Φ R be faces of G which lie to the right and left of the oriented edge p i p i respectively, see Figure 4. We will show that, for β small, a point of Φ R lies in the interior of i x D y p i Φ L p i x Φ R c x p i p i Figure 4.

10 10 NICHOLAS BARVINOK AND MOHAMMAD GHOMI Φ L. So C β,f is not simple. Since G admits only finitely many cut forests, this will complete the proof. To start, fix λ > 0 so small that (7) p i p i, p i c i λ p i p i. Let x be an interior point of p i p i such that (8) x p i λ/2, and note that x does not depend on β. Next set y := x + β x J(x c x ), r := β x λ/2, and let D be the disk of radius r centered at y. We will show that for β small: y x R < r, and D Φ L. Thus x R int(d ) int(φ L ), as desired. By the triangle inequality, y x R (x R x L) (y x) + (y x) (y x L). By Lemma 4.2 we may choose β so small that c x c x < λ/4. Then, by (5), (x R x L) (y x) = β x J(x c x ) β x J(x c x ) = β x c x c x < β x λ/4. Set δ := diam(p ). By Proposition 3.5, we can also make sure that (y x) (y x L) (λ/(4δ)) y x = (λ/(4δ))β x x c x β x λ/4, since c x P and therefore x c x δ. The last three displayed expressions yield that y x R < β xλ/2 = r, as claimed. As β 0, we have y x and r 0. Thus, for β small, D Φ R Φ L. Since y lies on the left side of the oriented line p i p i passing through p i and p i, it follows that y Φ L. So it remains to check that dist(y, p i p i ) r. By definition, c x = c pi = c i. Thus, by (7) and (8), So x c x, p i p i = x p i, p i p i + p i c i, p i p i (λ/2) p i p i. cos( c x xp i ) = x c x, p i p i x c x p i p i λ/2 x c x = β xλ/2 β x x c x = r y x, which yields dist(y, p i p i ) = sin( bxp i ) y x = cos( c xxp i ) y x r, and completes the proof. 5. A Non-monotone Convex Subdivision of the Equilateral Triangle A convex subdivision of a convex polygon is weighted if the corresponding graph G is weighted, as defined in Section 3. Further the subdivision, or its graph G, is non-monotone provided that G admits no monotone cut forests, as defined in Section 4. In this section we show: Theorem 5.1. Every equilateral triangle admits a non-monotone weighted convex subdivision.

11 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 11 One such subdivision, with over 500 vertices, is proposed in [20], although we have not verified it, since it is very complex. Our example has only 46 vertices, and is constructed via a different approach. However, as in [20], a key feature of our construction is a spiral to force the cut forest down a prescribed path Figure 5. For concreteness, let P be the equilateral triangle with vertices (0, 0), (100, 0), and (50, 50 3). The edge graph G of our subdivision of P is depicted in Figure 5. The exact coordinates of the vertices p i of G are listed in Table 1. We have also provided a Mathematica package [3] containing this data which may be used to check the claims below. Note that G contains a counterclockwise spiral S which starts at p 6 and ends at p 3. The critical vertices of G are p 1,..., p 25, which form S, the vortex p 5 of S, and the quadrilateral Q := p 1 p 2 p 3 p 4. The angles between the edges which meet at these vertices are all important as well. Other vertices are positioned just to ensure that the faces of G are convex. The main properties of G are described in the following three lemmas, which are trivial to check [3]. Lemma 5.2. For any vertex p i of S, with 4 < i < 25, p i+1 is the only vertex x adjacent to p i such that xp i p 5 π/2; if i = 4, then the only such vertex is x = p 3, and if i = 25, then the only such vertex is x = p 4.

12 12 NICHOLAS BARVINOK AND MOHAMMAD GHOMI 1 : ( , ) 2 : ( , ) 3 : ( , ) 4 : ( , ) 5 : ( , 20) 6 : ( , ) 7 : ( , ) 8 : ( , ) 9 : (56.942, ) 10 : ( , ) 11 : ( , ) 12 : ( , ) 13 : ( , ) 14 : ( , ) 15 : ( , ) 16 : ( , ) 17 : ( , ) 18 : ( , ) 19 : ( , ) 20 : ( , ) 21 : ( , ) 22 : ( , ) 23 : ( , ) 24 : ( , ) 25 : ( , ) 26 : ( , ) 27 : ( , ) 28 : ( , ) 29 : ( , ) 30 : ( , ) 31 : ( , ) 32 : ( , ) 33 : ( , ) 34 : ( , ) 35 : ( , ) 36 : (100, 0) 37 : ( , ) 38 : ( , ) 39 : ( , ) 40 : ( , ) 41 : ( , 65.7) 42 : (50, 50 3) 43 : ( , 65.7) 44 : ( , ) 45 : (0, 0) 46 : ( , ) Table 1. The interior angles of Q also satisfy the following obtuseness properties: Lemma 5.3. The only vertex x adjacent to p 3 such that p 4 p 3 x π/2 is p 2, the only vertex x adjacent to p 2 such that p 3 p 2 x π/2 is p 1, and the only vertex x adjacent to p 1 such that p 2 p 1 x π/2 is p 4. A path Γ in G, starting at some vertex p i, is called radially monotone with respect to p i if the distances of successive vertices of Γ from p i grow monotonically. Lemma 5.4. No path in G which is radially monotone with respect to a vertex of Q can reach p 5. Proof. Suppose that such a path, say Γ, exists, starting at a vertex p i of Q. Then the only choice for the penultimate vertex of Γ is p 6 ; because, as one may visually inspect, the only adjacent vertex x of p 5 such that x p i < p 5 p i for any i 4 is p 6. Next note that p 6 has only two adjacent vertices, p 7 and p 22, other than p 5. However, as trivial computations show, p 7 p i > p 6 p i and p 22 p i > p 6 p i for all i 4. So Γ cannot exist. Next we describe the weights α i associated to p i. Let 0 < µ < 1. Set w i := µ i for 1 i 5; w i := 0 for i = 36, 42, 45 (which correspond to the vertices of P ); and w i := µ 6 otherwise. Then set α i := w i / i w i. Note that as µ 0 α i α 5 α 4 α 3 α 2 α 1, for i 6. The main property of α i is the following fairly obvious fact: Lemma 5.5. Let X be a subset of interior vertices of G, c be the center of mass of X with respect to α i, and p j be the element of X with the smallest index. Suppose that j 5. Then c p j as µ 0. Furthermore if p k is the vertex of X with the next smallest index, and k 5, then p j c p k π.

13 PSEUDO-EDGE UNFOLDINGS OF CONVEX POLYHEDRA 13 For the rest of this section we fix a monotone cut forest F of G with respect to α i. The last observation, together with the monotonicity condition (3) yields: Lemma 5.6. Let p i be an interior vertex of G, and p j be the descendant of p i of smallest index, with respect to F. If j 5, then c i p j as µ 0. Furthermore, if p j p i, then we also have p i p ip j π/2, for sufficiently small µ. Proof. By Lemma 5.5, c i p j, as µ 0. When p j p i, p i p ip j is well-defined, and therefore so is p i p ic i for µ small. Furthermore we have p i p ic i p i p ip j. By (3) p i p ic i > π/2. So p i p ip j π/2 for µ sufficiently small. We also need a subtle variation of the last observation: Lemma 5.7. Let p i be an interior vertex of G, and p j be the strict descendant of p i of smallest index, with respect to F. If i < j 5, then p i p ip j π/2, for sufficiently small µ. Proof. By Lemma 5.5, p i c i p j π as µ 0. So (p i c i )/ p i c i converges to to (c i p j )/ c i p j. Furthermore, c i p i, by Lemma 5.5. Thus (p i c i )/ p i c i converges to (p i p j )/ p i p j. Hence p i p ic i p i p ip j, which completes the proof, since p i p ic i > π/2 by (3). Recall that, once we fix a cut forest of G, then each interior vertex of G may be connected to P by a unique ancestral path. Lemma 5.8. Let Γ be the ancestral path of p 5 with respect to F. Suppose that p 5 has no descendant in Q. Then, assuming µ is sufficiently small, Γ must contain a subpath in S ending at p 4. Proof. Γ must contain a vertex of S, because all adjacent vertices of p 5 lie on S. Let p i be the first vertex of S in Γ. Then, by Lemma 5.6, p i p ip 5 π/2, assuming µ is sufficiently small. Therefore, by Lemma 5.2, p i = p i+1. Similarly, p i+1 = p i+2, and thus Γ is forced to travel along S until it reaches p 4. Now we are ready to prove the main result of this section: Proof of Theorem 5.1. We claim that the subdivision of the triangle P given by the edge graph G and weights α i described above is non-monotone, for sufficiently small µ. Suppose, towards a contradiction, that G admits a monotone cut forest F for arbitrarily small µ. There are two cases to consider: either, with respect to F, p 5 has a descendant in Q or not. If p 5 has no descendant in Q, then, by Lemma 5.8, its ancestral path must connect p 5 to S and then travel along S until it reaches p 4. By Lemma 5.7, p 4 p 4p 5 π/2. Hence, by Lemma 5.3, p 4 = p 3. Similarly, Lemmas 5.7 and 5.3 yield that p 3 = p 2, p 2 = p 1 and p 1 = p 4. So F contains a loop, which is a contradiction. If p 5 has a descendant in Q, let p j be the one with smallest index. Then the ancestral path Γ j of p j passes through p 5. Let Γ j5 be the portion of Γ j connecting p j and p 5. By Lemma 5.6, for any vertex p i of Γ j5, c i p j as µ 0. Further, by the monotonicity condition (4), p i c i > p i c i. Thus, assuming µ is small,

14 14 NICHOLAS BARVINOK AND MOHAMMAD GHOMI p i p j > p i p j. So Γ j5 is radially monotone with respect to p j, which is not possible by Lemma 5.4. Hence, again we reach a contradiction. Theorem 5.1 together with Theorem 4.1 now immediately yield: Corollary 5.9. There exists a convex cap C over the equilateral triangle with a pseudo-edge graph with respect to which C is not unfoldable. Further, the total curvature of C may be arbitrarily small. 6. Proof of Theorem 1.1 We need only one more observation. A simple arc Γ in a topological space X is the image of a continuous mapping γ : [a, b] X, which is one-to-one on (a, b). We say Γ is a loop provided that γ(a) = γ(b). Lemma 6.1. Let E S 2 be an embedded graph which is isomorphic to the edge graph of a tetrahedron. Suppose there exists a simple arc Γ i in each face Φ i of E whose end points are distinct vertices of Φ i, and whose interior lies in the interior of Φ i. Then Γ := i Γ i contains a loop. Proof. Since E has only 4 vertices, we may assume that one of them, say v, is common to 3 different arcs Γ i, for otherwise Γ must contain a loop. Then all the vertices of the face Φ of E which is opposite to v belong to Γ. Consequently the arc in Φ forms a loop together with a pair of other arcs. Now we are ready to prove the main result of this work: Proof of Theorem 1.1. Let C be the convex cap over the equilateral triangle given by Corollary 5.9. By Lemma 3.1 we may assume that k(c) is so small that the total angles of C at each of its boundary vertices is less than 2π/3. Let C i, i = 1,..., 4, be congruent copies of C positioned over the faces of a regular tetrahedron, K := i C i, and E be the union of the pseudo-edges E i of C i. We claim that K is not unfoldable with respect to E. To see this suppose that T is a spanning tree of E, and let F i be the closure of the restriction of T to the interior of C i. Then F i will be a cut forest for E i provided that each connected component of F i intersects at most one vertex of C i. Thus, by Lemma 6.1, F j forms a spanning forest of E j for some 1 j 4. Consequently, the unfolding of C j with respect to F j is not simple. Hence the unfolding of K with respect to T is not simple, which completes the proof. Acknowledgements Clearly the authors are indebted to A. Tarasov for important investigations [20] which prompted and informed this work. Thanks also to J. O Rourke for his interest and useful comments on earlier drafts of this paper. Parts of this work were completed while the first named author participated in the REU program in the School of Math at Georgia Tech in the Summer of 2017.

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16 16 NICHOLAS BARVINOK AND MOHAMMAD GHOMI Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA address: URL: