A solution to some problems of Conway and Guy on monostable polyhedra
AA SOLUTION TO SOME PROBLEMS OF CONWAY AND GUYON MONOSTABLE POLYHEDRA
ZSOLT L ´ANGI
Abstract.
A convex polyhedron is called monostable if it can rest in stableposition only on one of its faces. The aim of this paper is to investigate threequestions of Conway, regarding monostable polyhedra, which first appearedi a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy,
Stabilityof polyhedra (J.H. Conway and R.K. Guy) , SIAM Rev. (1969), 78-82).In this note we answer two of these problems and make a conjecture aboutthe third one. The main tool of our proof is a general theorem describingapproximations of smooth convex bodies by convex polyhedra in terms of theirstatic equilibrium points. As another application of this theorem, we prove theexistence of a convex polyhedron with only one stable and one unstable point. Introduction
The study of static equilibrium points of convex bodies started with the workof Archimedes [22], and has been continued throughout the history of science invarious disciplines: from geophysics and geology [16, 29] leading to examinationof the possible existence of water on Mars [30], to robotics and manufacturing[31, 4] to biology and medicine [1, 11, 17]. In modern times, the mathematicalaspects of this concept was started by a problem of Conway and Guy [5] in 1966who conjectured that there is no homogeneous tetrahedron which can stand inrest only on one of its faces when placed on a horizontal plane, but there is ahomogeneous convex polyhedron with the same property. These two questions wereanswered by Goldberg and Guy in [20] in 1969, respectively (for a more detailedproof of the first problem, see [7]), who called the convex polyhedra satisfying thisproperty monostable or unistable . In addition, in [20] Guy presented some problemsregarding monostable polyhedra, stating that three of them are due to Conway (forsimilar statements in the literature, see e.g. [6, 8, 18]). These three questionsappear also in the problem collection of Croft, Falconer and Guy [6] as ProblemB12. The aim of our paper is to examine these problems. Problem 1.
Can a monostable polyhedron in the Euclidean 3-space R have an n -fold axis of symmetry for n > Mathematics Subject Classification.
Key words and phrases. monostable, unistable, equilibrium point, G¨omb¨oc, polyhedralapproximation.The author is supported by the National Research, Development and Innovation Office, NKFI,K-119245, the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and theBME IE-VIZ TKP2020 and ´UNKP-20-5 New National Excellence Programs by the Ministry ofInnovation and Technology. a r X i v : . [ m a t h . M G ] N ov Z. L´ANGI
Before the next problem, recall that the girth of a convex body in R is theminimum perimeter of an orthogonal projection of the body onto a plane [18]. Problem 2.
What is the smallest possible ratio of diameter to girth for a monos-table polyhedron?
Problem 3.
What is the set of convex bodies uniformly approximable by monos-table polyhedra, and does this contain the sphere?It is worth noting that, according to Guy [20], Conway showed that no body ofrevolution can be monostable, and also that the polyhedron constructed in [20] hasa 2-fold rotational symmetry. Problems 1-3 appear also in the problem collection ofof Klamkin [25], and Problem 1 and some other problems for monostable polyhedraappear in a 1968 collection of geometry problems of Shephard [28], who describedthese objects as ‘a remarkable class of convex polyhedra’ whose properties ‘it wouldprobably be very rewarding and interesting to make a study of’.Before stating our main result, we collect some related results and problemsfrom the literature. Here, we should first mention from [20] the problem of findingthe minimal dimension d in which a d -simplex can be monostable. This problemhave been investigated by Dawson et al. [7, 10, 9, 8], who proved that there is nomonostable d -simplex if d ≤ R , the original construction of Guy [20] with 19 faces (attributedalso to Conway) was modified by Bezdek [3] to obtain a monostable polyhedronwith 18 faces, while a computer-aided search by Reshetov [26] yields a monostablepolyhedron with 14 faces. In [23], Heppes constructed a homogeneous tetrahedronin R with the property that putting it on a horizontal plane with a suitable face,it rolls twice before finding a stable position. Another interesting convex body isfound by Dumitrescu and T´oth [18], who constructed a convex polyhedron P withthe property that after placing it on a horizontal plane with a suitable face, it coversan arbitrarily large distance while rolling until it finds a stable position. Finally, weremark that a systematic study of the equilibrium properties of convex polyhedrawas started in [13].Our main result is the following, where d H ( · , · ) and B denotes the Hausdorffdistance of convex bodies, and the closed unit ball in R centered at the origin o ,respectively. Theorem 1.
For any n ≥ , n ∈ Z and ε > there is a homogeneous monostablepolyhedron P such that P has an n -fold rotational symmetry and d H ( P, B ) < ε . Theorem 1 answers Problem 1, and also the case of a sphere in Problem 3. Inaddition, from it we may deduce Corollary 1 (for the second part, see also [18]).This solves Problem 2. Here, for any convex body K ⊂ R , we denote by diam( K )and g ( K ) the diameter and the girth of K , respectively. Corollary 1.
For any ε > there is a monostable polyhedron P with diam( P ) g ( P ) < π + ε . Furthermore, we have diam( K ) g ( K ) ≥ π for any convex body K ⊂ R . The proof of Theorem 1 is based on a general theorem on approximation ofsmooth convex bodies by convex polytopes. Before stating it, we briefly introduce
ONOSTABLE POLYHEDRA 3 some elementary concepts regarding their equilibrium properties. Let K ⊂ R be asmooth centered convex body, and let δ K : bd( K ) → R be the Euclidean distancefunction measured from o , where bd( K ) denotes the boundary of K . The criticalpoints of δ K are called equilibrium points of K . To avoid degeneracy, it is usuallyassumed that δ K is a Morse function; i.e. it has finitely many critical points, bd( K )is twice continuously differentiable at least in a neighborhood of each critical point,and at each such point the Hessian of δ K is nondegenerate [34]. Depending onthe number of negative eigenvalues of the Hessian, we distinguish between stable , unstable and saddle-type equilibrium points, corresponding to the local minima,maxima and saddle points of δ K , respectively. The Poincar´e-Hopf Theorem impliesthat under these conditions, the numbers S , U and H of the stable, unstable andsaddle points of K , respectively, satisfy the equation S − H + U = 2.Answering a conjecture of Arnold, Domokos and V´arkonyi [32] proved that thereis a homogeneous convex body with only one stable and one unstable point. Theycalled the body they contructed ‘G¨omb¨oc’ (for more information, see [21]). Inaddition to the existence of G¨omb¨oc, in their paper [32] Domokos and V´arkonyiproved the existence of a convex body with S stable and U unstable equilibriumpoints for any S, U ≥
1. This investigation was extended in [15] to the combinatorialequivalence classes defined by the Morse-Smale complexes of ρ K , and in [12] fortransitions between these classes. Based on these results, for any S, U ≥ S, U ) c as the family of smooth convex bodies K having S stable and U unstable equilibrium points, where K has no degenerate equilibrium point, andat each such point bd( K ) has a positive Gaussian curvature. We define the class( S, U ) p analogously for convex polyhedra, where stable and unstable points of aconvex polyhedron are defined formally in Section 2. Our theorem is the following,where we call a convex body centered if its center of mass is the origin o . Theorem 2.
Let ε > , S, U ≥ be arbitrary, and let G be any subgroup ofthe orthogonal group O (3) . Then for any centered, G -invariant convex body K ∈ ( S, U ) c , there is a centered G -invariant convex polyhedron P ∈ ( S, U ) p such that d H ( K, P ) < ε . Here we note that the fact that any nondegenerate convex polyhedron can beapproximated arbitrarily well by a smooth convex body with the same number ofequilibrium points is regarded as ‘folklore’ (we use a simple argument to show itin Section 2). On the other hand, it is shown in [14] that any sufficiently fineapproximation of a smooth convex body K by a convex polyhedron P , using anequidistant partition of the parameter range of the boundary of K , has strictly morestable, unstable and saddle points in general than the corresponding quantities for K .Even though the convex body constructed in [32] is not C -class at its two equi-librium points, in [15] it is shown that class (1 , c is not empty. Thus, Theorem 2readily implies the existence of a polyhedron in class (1 , p . Corollary 2.
There is a convex polyhedron with a unique stable and a uniqueunstable point.
Furthermore, we remark that the elegant construction in the paper [18] of Du-mitrescu and T´oth yields an inhomogeneous monostable convex polyhedron arbi-trarily close to a sphere. Nevertheless, we must add that dropping the requirement
Z. L´ANGI of uniform density may significantly change the equilibrium properties of a convexbody. To show it we recall the construction of Conway (see [8]) of an inhomoge-neous monostable tetrahedron in R , and observe that spheres with inhomogeneousdensity, as also roly-poly toys, yield trivial solutions to Arnold’s conjecture.For completeness, we also recall the remarkable result of Zamfirescu [33] statingthat a typical convex body (in Baire category sense) has infinitely many equilibriumpoints, and note that critical points of the distance function from another point areexamined in Riemannian manifolds, e.g. in [2, 19, 24].In Section 2, we introduce our notation and collect the necessary tools for provingTheorems 1 and 2, including more precise definitions for some of the conceptsmentioned in Section 1. In Section 3 we prove Theorem 1 and Corollary 1. InSection 4 we prove Theorem 2. Finally, in Section 5 we collect some additionalremarks and ask some open questions.2. Preliminaries
In the paper, for any p, q ∈ R , we denote by [ p, q ] the closed segment with end-points p, q , and by | p | the Euclidean norm of p . We denote the closed 3-dimensionalunit ball centered at the origin o by B , and its boundary by S . Furthermore, forany set S ⊂ R we let conv( S ) denote the convex hull of S . By a convex body wemean a compact, convex set with nonempty interior.Let K ⊂ R be a convex body. The center of mass c ( K ) of K is defined by thefraction c ( K ) = K ) (cid:82) x ∈ K x dv , where v denotes 3-dimensional Lebesgue measure.We remark that the integral in this definition is called the first moment of K , andnote that we clearly have c ( K ) ∈ int( K ) for any convex body K . If q ∈ bd( K )satisfies the property that the plane through q and orthogonal to the vector q − c ( K )supports K , then we say that q is an equilibrium point of K . Here, if K is smooth,then the equilibrium points of K coincide with the critical points of the Euclideandistance function measured from c ( K ) and restricted to bd( K ). We remark that aconvex body K ⊂ R is called smooth if for any boundary point x of K there is aunique supporting plane of K at q ; this property coincides with the property thatbd( K ) is a C -class submanifold of R (cf. [27]).We define nondegenerate equilibrium points only in two special cases. If K issmooth, q ∈ bd( K ) is an equilibrium point of K with a C -class neighborhood inbd( K ), and the Hessian of the Euclidean distance function on bd( K ), measuredfrom c ( K ), is nondegenerate, we say that q is nondegenerate . In this case q iscalled a stable, saddle-type or unstable point of K if the number of the negativeeigenvalues of the Hessian at q is 0 , K is a convex polyhedron in R , and q ∈ bd( K ) is an equilibrium point of K . Then there is a unique vertex, edge or face of K that contains q in its relativeinterior. Let F denote this face, and let H be the supporting plane of K through q that is perpendicular to q − c ( K ). Observe that F ⊂ K ∩ H . We say that q is nondegenerate if F = K ∩ H . In this case we call q a stable, saddle-type or unstable point of K if the dimension of F is 2 , K is called nondegenerate if it has only finitely manyequilibrium points, and each such point is nondegenerate. We note that in theabove definitions, we may replace the center of mass of K by any fixed reference ONOSTABLE POLYHEDRA 5 point c ∈ K . In this case we write about equilibrium points relative to c . Weemphasize that in the paper, unless it is stated otherwise, if the reference point isnot specified, then it is meant to be the center of mass of the body.Let K ∈ R be a nondegenerate smooth convex body with S stable, H saddle-type and U unstable equilibrium points. Using a standard convolution technique,we may assume that K has a C ∞ -class boundary, and hence, by the Poincar´e-HopfTheorem, we have S − H + U = 2 [15]. We show that the same holds if K is anondegenerate convex polyhedron. Indeed, let τ > K ( τ ) = ( K ÷ ( τ B )) + ( τ B ), where ÷ denotes Minkowski difference and + denotesMinkowski addition [27]. Then for any τ > K ( τ ) is a smooth nondegenerateconvex body having the same numbers of stable, saddle-type and unstable pointsrelative to c ( K ); hence, we may apply the Poincar´e-Hopf Theorem for K ( τ ) (here wenote that by Lemma 2 the same property holds relative to c ( K ( τ )) as well). Thus,for any nondegenerate convex body, the numbers of stable and unstable pointsdetermine the number of saddle-type points. We define class ( S, U ) c as the familyof nondegenerate, smooth convex bodies K ⊆ R with S stable, U unstable pointswith the additional assumption that at each equilibrium point of K , the principalcurvatures of bd( K ) are positive. Similarly, by ( S, U ) p we mean the family ofnondegenerate convex polyhedra with S stable and U unstable points. Observethat if K is nondegenerate, the point of bd( K ) closest to or farthest from c ( K ) isnecessarily a stable or unstable point, respectively, implying that the numbers S, U in the above symbol are necessarily positive.For the following remark, see Lemma 7 from [15].
Remark 1.
Let K ∈ ( S, U ) Ec and for any equilibrium point q of K , let V q be anarbitrary compact neighborhood of q containing no other equilibrium point of K .Then c ( K ) has an open neighborhood U such that for any x ∈ U , K has S stableand U unstable points relative to x , and for any equilibrium point q of K relativeto c ( K ), V q contains exactly one equilibrium point of K relative to x , and the typeof this point is the same as the type of q .For Remark 2, see the paragraph in [14] after Definition 2. Remark 2.
Let q be an equilibrium point of a centered convex body K in ( S, U ) c for some S, U ≥
1. Let | q | = ρ , and let κ , κ denote the principal curvatures ofbd( K ) at q . Then κ , κ (cid:54) = ρ . Furthermore, 0 ≤ κ , κ < ρ if and only if q is astable point, κ , κ > ρ if and only if q is an unstable point, and 0 < min { κ , κ } < ρ < max { κ , κ } if and only if q is a saddle-type equilibrium point. Lemma 1.
The symmetry group of any nondegenerate convex body K is finite.Proof. Let K be a nondegenerate convex body with symmetry group G . Withoutloss of generality, assume that K is centered, i.e. c ( K ) = o . Since c ( K ) is clearly afixed point of any symmetry in G , we have that G is a subgroup of the orthogonalgroup O (3). Clearly, G is closed in O (3), and thus, it is a Lie group embedded in O (3) by Cartan’s Closed Subgroup Theorem. On the other hand, the Lie subgroupsof O (3) are well known, and in particular we have that if G is infinite, then itcontains, up to conjugacy, SO (2) as a subgroup. In other words, K is rotationallysymmetric. Thus, by nondegeneracy, K has exactly one stable and one unstable Z. L´ANGI equilibrium point. But this property contradicts Conway’s result mentioned inSection 1 that no rotationally symmetric convex body is monostable. (cid:3)
We finish Section 2 with two lemmas and two remarks, where X (cid:52) Y denotes thesymmetric difference of the sets X, Y . Lemma 2.
Let K ( τ ) ⊂ R be a -parameter family of convex bodies, where τ ∈ [0 , τ ] for some τ > . For any τ ∈ [0 , τ ] , let c ( τ ) denote the center of mass of K ( τ ) , and let K = K (0) and c = c (0) . Assume that for some C > and m > , vol( K ( τ ) (cid:52) K ) ≤ Cτ m holds for any sufficiently small value of τ . Then there issome C (cid:48) > such that | c ( τ ) − c | ≤ C (cid:48) τ m holds for any sufficiently small value of τ .Proof. Without loss of generality, we may assume that K ( τ ) ⊆ r B if τ is suffi-ciently small. By definition, c ( τ ) = (cid:82) x ∈ K ( τ ) x dv vol( K ( τ )) . On the other hand, by the condi-tions, we have | vol( K ( τ )) − vol( K ) | ≤ Cτ m , and | (cid:82) x ∈ K ( τ ) x dλ − (cid:82) x ∈ K x dv | ≤ rCτ m for all sufficiently small values of τ . From these inequalities and the fact thatvol( K ) >
0, the assertion readily follows. (cid:3)
Lemma 3.
Let p ∈ int( B ) ⊂ R and q ∈ S such that p , q and o are not collinear,and let L be a line through p such that L does not separate o and q . Furthermore,if A denotes the convex angular region with q ∈ A and bounded by a half line of L starting at p , and the half line starting at p and containing o , then assume thatthe angle of A is obtuse. Then there is a convex polygon Q ⊂ B with vertices o, x = q, x , . . . , x k = p in cyclic order in bd( Q ) such that x i − x i o ∠ > π for allvalues of i , and L supports Q . q=xx' x' k p o Figure 1.
The construction of the points x (cid:48) i in the proof ofLemma 3. The dotted curves indicate arcs in the Thales circlesof the segments [ o, x (cid:48) i ].We remark that the conditions in Lemma 3 imply that the Euclidean distancefunction x (cid:55)→ | x | , x ∈ R strictly decreases along the curve (cid:83) ki =1 [ x i − , x i ] from q to p . Proof.
Without loss of generality, we may assume that q = (1 ,
0) and the y -coordinate of p is positive. Set poq ∠ = β ∈ (0 , π ), and choose an arbitrary positiveinteger k . For any i = 0 , , . . . , k , define the point x (cid:48) i = (cid:16) r i cos iβk , r i sin iβk (cid:17) , where r i = cos i βk . Then x (cid:48) = q , and x (cid:48) i is on the Thales circle of the segment [0 , x (cid:48) i − ], and ONOSTABLE POLYHEDRA 7 thus, x (cid:48) i − x (cid:48) i o ∠ = π (cf. Figure 1) for all i = 1 , , . . . , k . Using elementary calculus,we obtain that lim k →∞ cos k βk = 1, which yields that there is some value of k suchthat | x (cid:48) k | > | p | . Since x (cid:48) k and p are on the same half line, we may decrease the valuesof r i for i = 1 , , . . . , k slightly such that for the points x i obtained in this way theconvex polygon Q = conv { o, x , x , . . . , x k } satisfies the required conditions apartfrom the one for L . Now, if L supports Q , we are done. On the other hand, if L does not support Q , then we may take the polygon obtained as the intersection of Q and the closed half plane bounded by L and containing o in its interior. (cid:3) Remark 3.
Let a, b >
0, where a (cid:54) = b , and let E ⊂ R be the ellipse with equation x a + y b ≤
1. Then, for any δ > ε > K ⊂ R is aplane convex body satisfying E ⊆ K ⊆ (1 + δ ) E , and the vector w is perpendicularto a supporting line of K through w ∈ bd( K ), then the angle between w and the x -axis or the y -axis is at most δ . Remark 4.
Let f, g be two real functions defined in a neighborhood of a ∈ R . If f, g are both locally strictly increasing (resp., decreasing) at a , then so are min { f, g } and max { f, g } .Finally, we remark that in the proof of Theorem 2, we use ideas also from[15, 12, 18]. 3. Proofs of Theorem 1 and Corollary 1
First, we show how Theorem 2 implies Theorem 1.Let n ≥ ε > K ∈ (1 , m ) c forsome value of m with n -fold rotational symmetry and satisfying d H ( K, B ) ≤ ε .Let P be a regular n -gon inscribed in a fixed circle C on B parallel to, but notcontained in the ( x, y )-plane. Let the vertices of P be p i , i = 1 , , . . . , n . Let Q ( ε ) =conv (cid:0) B ∪ { (1 + ε ) p , . . . , (1 + ε ) p n } (cid:1) . Then Q ( ε ) is the union of B and n cones C i , i = 1 , , . . . , n , with spherical circles centered at the points p i as directrixes. Bysymmetry, the center of mass c of Q ( ε ) is on the z -axis, and by the Thales Theoremimplies and Lemma 2, its distance from o is of magnitude O ( ε ). Thus, the points(1 + ε ) p i are equilibrium points of Q ( ε ) if ε is sufficiently small. Furthermore, wehave c (cid:54) = o . On one hand, from this we have that there are exactly two equilibriumpoints of Q ( ε ) on S , namely the points (0 , ,
1) and (0 , , − Q ( ε ) has exactly one equilibrium point on each cone C i apart from itsvertex; this point is a saddle point in the relative interior of a generating segmentof C i (cf. Figure 2).Now, we set Q (cid:48) ( ε ) = ( Q ( ε ) ÷ ( τ B )) + ( τ B ), where τ > ε . Then Q (cid:48) ( ε ) is a smooth convex body which has 1 stable, n saddle-type and ( n +1) unstable points by Lemma 2. To guarantee that the body has positive principalcurvatures at each equilibrium point, we may replace the generating segments ofthe cones by circular arcs of radius R >
0, where R is negligible compared to τ .The obtained convex body K ( ε ) ∈ (1 , n + 1) c satisfies the required conditions.Finally, we prove Corollary 1. Clearly, diam( B ) = 2 and g ( B ) = 2 π , andhence, the first statement follows from the continuity of diameter and girth with Z. L´ANGI o q i z-axis Figure 2.
An illustration for the proof of Theorem 1.respect to Hausdorff distance. On the other hand, let K ⊂ R be a convex body,and let w ( · ) and perim( · ) denote mean width and perimeter, respectively. Then, forany projection M of K , we have ≤ w ( M ) ≤ diam( M ) ≤ diam( K ). On the otherhand, it is well known that w ( M ) = perim( M ) π , implying that diam( K )perim( M ) ≥ π . Fromthis, we readily obtain diam( K ) g ( K ) ≥ π .4. Proof of Theorem 2
First, observe that by Lemma 1, G is finite.We construct P by truncating K with finitely many suitably chosen planes; ormore precisely by taking its intersection with finitely many suitably chosen closedhalf spaces. We carry out the construction of P in three steps.In Step 1, we replace some small regions of bd( K ) by polyhedral regions disjointfrom all equilibrium points of K . These polyhedral regions will serve as ‘controllingregions’; that is, after constructing a polyhedron with S stable and U unstablepoints relative to o , we modify these regions to move back the center of mass ofthe constructed polyhedron to o . In Step 2, we truncate a neighborhood of eachequilibrium point to replace it by a polyhedral surface in such a way that eachpolyhedral surface contains exactly one equilibrium point relative to o , and thetype of this point is the same as the type of the corresponding equilibrium point of K . Finally, in Step 3 we truncate the remaining part of bd( K ) such that no newequilibrium point is created.In the proof, we denote by E the set of the equilibrium points of K , and forany point q ∈ bd( K ), we denote by H q the unique supporting plane of K at q .Observe that by the definition of ( S, U ) c , H q ∩ K = { q } for any q ∈ E , and set X = bd( K ) \ E . Finally, by F we denote the set of the fixed points of G , and notethat F is a linear subspace of R that contains the center of mass of any G -invariantconvex body. Step 1 .We distinguish two cases depending on dim( F ). ONOSTABLE POLYHEDRA 9
Case 1 , if F = R . By Carath´eodory’s theorem, there are points z , z , z , z ∈ X such that o ∈ conv { z , z , z , z } . Since X is open in bd K , we may choose thesepoints to satisfy o ∈ int conv { z , z , z , z } . By the definition of ( S, U ) c , we havethat H z i is disjoint from H q for any 1 ≤ i ≤ q ∈ E . We show that the z i s canbe chosen such that the planes H z i are pairwise distinct. Suppose for contradictionthat, say, three of these planes coincide. Without loss of generality, assume that H z = H z = H z , and denote this common plane by H . Then there are points z (cid:48) , z (cid:48) , z (cid:48) ∈ relbd( K ∩ H ) such that conv { z , z , z } ⊆ conv { z (cid:48) , z (cid:48) , z (cid:48) } . Now we mayreplace z (cid:48) and z (cid:48) by two points z (cid:48)(cid:48) , z (cid:48)(cid:48) / ∈ H such that z (cid:48)(cid:48) i is sufficiently close to z (cid:48) i for i = 2 ,
3. Then we have o ∈ int conv { z (cid:48) , z (cid:48)(cid:48) , z (cid:48)(cid:48) , z } , where no supporting plane of K contains three of the points. If a supporting plane of K contains two of these points,we may repeat the above procedure, and finally obtain some points w , . . . , w ∈ X such that o ∈ int conv { w , . . . , w } , and the sets H w i ∩ K are pairwise disjoint.Let δ >
0, and let us truncate K by planes H , . . . , H such that for all i s H i is parallel to H z i and it is at the distance δ from it in the direction of o .We denote the truncated convex body by K (cid:48) and its center of mass by c (cid:48) . ByRemark 1, if δ is sufficiently small, then K has S stable and U unstable equilibriumpoints relative to o (cid:48) , and each such equilibrium point is contained in bd( K (cid:48) ) \ ( (cid:83) i =1 H i ). Furthermore, if δ is sufficiently small, then for any point q ∈ H i ∩ bd( K )and any plane H supporting K (cid:48) at q , q is not perpendicular to H . Finally, since o ∈ int conv { w , . . . , w } , we may choose points w (cid:48) i ∈ relint( H i ∩ K (cid:48) ) such that c (cid:48) int conv { w (cid:48) , . . . , w (cid:48) } . For any w (cid:48) i , choose some convex n i -gon P i ⊂ relint( H i ∩ K (cid:48) ) such that the center of mass of P i is w (cid:48) i . Now we obtain the body K (cid:48)(cid:48) bytruncating K (cid:48) by n i planes almost parallel to H i such that for each i , every sideof P i is contained in one of the truncating planes, and we have P i = H i ∩ K (cid:48)(cid:48) . Wechoose the truncating planes such that the center of mass c (cid:48)(cid:48) of K (cid:48)(cid:48) satisfies c (cid:48)(cid:48) ∈ int conv { w (cid:48) , . . . , w (cid:48) } , and K (cid:48)(cid:48) has S stable and U unstable points on the smoothpart of its boundary, and no equilibrium point on the non-smooth part. Now, weset K = K (cid:48)(cid:48) − c (cid:48)(cid:48) , and q (cid:48) i = q i − c (cid:48)(cid:48) , P (cid:48) i = P i − c (cid:48)(cid:48) for all i s, and for some sufficientlysmall ¯ τ > C i ( τ i ) = conv( P (cid:48) i ∪ { (1 + τ i ) q (cid:48) i } ), τ i ∈ [0 , ¯ τ ], i = 1 , , ,
4. Furthermore, for later use, we set K = K − c (cid:48)(cid:48) , and callthe set X = K ∩ bd( K ) the non-truncated part of bd( K ).If ¯ τ is sufficiently small, then K ∪ (cid:83) i =1 C i ( τ i ) is convex for all values of theparameters τ i . Furthermore, the first moment of (cid:83) i = 1 C i ( τ i ) is (cid:80) i =1 α i τ i q (cid:48) i forsome suitable constants α i >
0, which implies that it is surjective in a neighborhoodof o . Thus, since K is centered, after we replace the non-truncated part of bd( K )by a polyhedral surface in Steps 2 and 3, we may choose values of the τ i s in sucha way that the sum of the first moment of (cid:83) i =1 C i ( τ i ) and of the first momentof the polyhedron P obtained after Step 3 is equal o . This makes the polyhedron P ∪ (cid:83) i = 1 C i ( τ i ) centered. Finally, we observe that by choosing sufficiently smallvalues of δ and ¯ τ , for all values of the parameters, no point of C i ( τ i ) is an equilibriumpoint of K relative to o . Case 2 , if F (cid:54) = R . In this case F is a plane or a line through o , or F = { o } .Consider the case that F is a plane. Then, by the properties of isometries, theorbit of any point p under G consists of p and its reflection about F . Let K F = F ∩ K , and observe that since K is symmetric about F , for any q ∈ bd( K ) H q is either disjoint from K F or q ∈ relbd( K F ). Thus, we may apply the argument in Case 1 for K F , and obtain some points z , z , z ∈ X ∩ K F such that o ∈ relint conv { z , z , z } and the planes H z i are pairwise disjoint. But then there aresome points z (cid:48) and z (cid:48)(cid:48) , sufficiently close to H z such that z (cid:48)(cid:48) is the reflected copyof z (cid:48) about L , o ∈ (cid:82) conv { z , z , z (cid:48) , z (cid:48)(cid:48) } , and the supporting planes at these pointsare pairwise disjoint. Clearly, the set { z , z , z (cid:48) , z (cid:48)(cid:48) } is G -invariant. From now on,we may apply the argument in Case 1. If F is a line, we may repeat the argumentin the previous paragraph. Finally, if F = { o } , then any G -invariant convex body(and in particular the convex polyhedron constructed in Steps 2 and 3) is centered.Thus, in this case we may skip Step 1.Based on the existence of the families C i ( τ i ), in Steps 2 and 3 all equilibriumpoints are meant to be relative to o . We denote by E the set of the equilibriumpoints of K . Step 2 .In this step we take all points q ∈ E , and truncate neighborhoods of them inbd( K ) simultaneously for all points in the orbit of q . Here we observe that the orbitof an equilibrium point consists of equilibrium points. We carry out the truncationsin such a way that the regions truncated in Step 1 or Step 2 are pairwise disjoint.We denote the convex body obtained in this step by K , and set X = bd( K ) ∩ K .We construct K in such a way that for any point p ∈ X there is no supportingplane H of K through p which contains an equilibrium point of K .Consider some q ∈ E . Without loss of generality, we may assume that q =(0 , , ρ ) for some ρ >
0, and denote by e x , e y , and e z the vectors of the standardorthonormal basis. With a little abuse of notation, for any p ∈ bd( K ), we denoteby H p the unique supporting plane of K at p . Case 1 , the stabilizer of q in G is the identity; i.e. q not fixed under any elementof G other than the identity. Subcase 1.1 , q is a stable point of K . In this case we truncate K by a plane H (cid:48) q parallel to, and sufficiently close to H q . Then we truncate K by finitely manyadditional planes such that any point of H (cid:48) q ∩ bd( K ) is truncated by at least oneof them, and for any point p of the non-truncated part X of bd( K ) there is nosupporting plane H of K through p which contains an equilibrium point of K relative to o . Subcase 1.2 , q is a saddle-type equilibrium point. Note that by Remark 2, q isnot an umbilic point of bd( K ), and its principal curvatures κ < κ satisfy theinequalities 0 < κ < ρ < κ .Without loss of generality, we may assume that the sectional curvature of bd( K )in the ( x, z )-plane is κ , and in the ( y, z )-plane it is κ . For any τ >
0, let K ( τ )denote the set of points of K with z -coordinates at least ρ − τ , and observe thatby the fact that κ > κ >
0, for any ε > τ > K ( τ ) iscontained in the neighborhood of q of radius ε . For any { i, j } ⊂ { x, y, z } , let H ij denote the ( i, j ) coordinate plane, and proj ij denote the orthogonal projection of R onto H ij .For any η >
0, let C ( η ) be the set of the points of the circular disk y + ( z − ρ + η ) ≤ η in H yz whose z -coordinates are at least ρ − τ . Then, since bd( K ) ONOSTABLE POLYHEDRA 11 is C -class in a neighborhood of q , we have that for any η , η > ρ < η < κ < η , if τ is sufficiently small, then C ( η ) ⊆ proj yz ( K ( τ )) ⊆ C ( η )holds. Since proj yz ( K ) is convex, relbd(proj yz ( K )) has exactly two points withtheir z -coordinates equal to ρ − τ . Let these points be q − = (0 , σ − , ρ − τ ) and q + = (0 , σ + , ρ − τ ) such that σ − < < σ + . Then there are some supportinglines L − , L + of proj yz ( K ) passing through q − and q + , respectively. Clearly, for i ∈ {− , + } , L i is the orthogonal projection of some supporting plane H i of K onto H yz . Let r i be a point of L i , on the open half line starting at q i such that theline through [ o, q i ] do not separate q and r i . Then, for any fixed values of η and η and sufficiently small value of τ , the angles oq i r i ∠ are obtuse. Now we choosesome sufficiently small value of ζ >
0, and define q i (cid:48) = ζe z + q i , r i (cid:48) = ζe z + r i , L i (cid:48) = ζe z + L i , H i (cid:48) = ζe z + H i and q (cid:48) = − ζe z + q . Then we may assume that | q i (cid:48) | < | q (cid:48) | , the angles oq i (cid:48) r i (cid:48) ∠ are obtuse, and the planes H i (cid:48) are disjoint from K .Thus, by Lemma 3, for i ∈ {− , + } , there is a polygonal curve Γ i in H yz , con-necting q (cid:48) to q i (cid:48) such that the Euclidean distance measured from the points of Γ i to o is strictly decreasing as we move from q (cid:48) to q i (cid:48) (see the remark after Lemma 3),Γ i is contained in relbd(conv(Γ i ∪ { o } )), and the latter set is supported by L i (cid:48) in H yz . Consider the closed, convex set C H ⊂ H yz bounded by Γ − ∪ Γ + , the halfline of L + (cid:48) starting at q + (cid:48) and not containing r + (cid:48) , and the half line of L −(cid:48) startingat q −(cid:48) and not containing r −(cid:48) , and set C = proj − yz ( C H ) ⊂ R . By the previousconsideration, C is an infinite convex cylinder with the properties that o ∈ int( C ), K \ C ⊆ K ( τ ), and the equilibrium points of C relative to o are q and two stablepoints on L + (cid:48) and L −(cid:48) , respectively. To construct K , we truncate K by C , andshow that, apart from the saddle point q (cid:48) , no new equilibrium point is created bythis truncation.Observe that by our construction, any new equilibrium point is a point ofbd( K ) ∩ bd( C ). Suppose that there is some equilibrium point q ∈ bd( K ) ∩ bd( C )of K ∩ C . To reach a contradiction, we identify H xy with R , and parametrizebd( K ) in a neighborhood of q as the graph of a function f : R → R and bd( C ) ina neighborhood of q (cid:48) as the graph of a function g : R → R . Note that by the non-degeneracy of q , for some value of φ > o has a neighborhood U ⊂ R such thatfor any w = ( x , y ) ∈ U whose angle with the x -axis is at most φ , | ( x, y, f ( x, y ) | is locally strictly increasing at w as a function of x if x > x <
0; furthermore, if the angle of w with the y -axis is at most φ , then | ( x, y, f ( x, y ) | is locally strictly decreasing as a function of y if y > y <
0. Note that by Remark 4, the same property holds formin {| ( x, y, f ( x, y )) | , | ( x, y, g ( x, y )) |} as well. Observe that since q is an equilibriumpoint of K ∩ C , it is an equilibrium point of the section of K ∩ C with the planethrough q parallel to H xy . Thus, by Remark 3, if τ > xy ( q ) with the x -axis or the y -axis is at most φ . But this contradictsour previous observation that at such a point min {| ( x, y, f ( x, y )) | , | ( x, y, g ( x, y )) |} is locally strictly increasing or decreasing parallel to the x - or the y -axis.Finally, to exclude the possibility that a support plane of K ∩ C through a pointin bd( K ) ∩ bd( C ) contains q (cid:48) , we truncate all points of bd( K ) ∩ bd( C ) by planes,not containing q (cid:48) , whose intersections with K ∩ C do not contain equilibrium point. Subcase 1.3 , q is an unstable point. In this case both principal curvatures κ , κ of bd( K ) at q satisfy κ , κ > ρ >
0, and thus, there is some constantmax (cid:110) κ , κ (cid:111) < η < ρ such that the ball ρ − ηρ q + η B contains a neighborhoodof q in bd( K ). We parametrize bd( K ) in a neighborhood of q as the graph { z = f ( x, y ) } of a function ( x, y ) (cid:55)→ f ( x, y ), and note that by nondegeneracy, thefunction | ( x, y, f ( x, y ) | is strictly decreasing in a neighborhood of (0 ,
0) as a functionof (cid:112) x + y .For any τ >
0, let K ( τ ) denote the set of points of K with z -coordinates at least ρ − τ , let H τ denote the plane with equation { z = ρ − τ } . Let τ > C centered at (0 , , ρ − τ ) which is contained in H τ ∩ int( η B ), and is disjoint from K . Let H be a plane supporting K at a pointof H τ such that its angle α with H τ is minimal among these supporting planes.Let H (cid:48) be the translate of H touching C such that H strictly separates o and H (cid:48) ,and let the intersection point of H (cid:48) and the z -axis be r . Consider the infinite cone C with apex r and base C , and observe that it contains K \ K ( τ ) in its interior.Now, let q (cid:48) = q − ζe z for some suitably small ζ >
0, and let Γ be a polygonalcurve connecting q (cid:48) to a point p ∈ C such that the plane of o, p, q (cid:48) contains Γ,Γ ⊂ relbd(conv(Γ ∪ { o } )), and the Euclidean distance function is strictly decreasingalong Γ from q (cid:48) to p . Let L p denote the closed half line in the line of [ r, p ] startingat p and not containing r , and let Γ (cid:48) = Γ ∪ L p . Let m ≥ i = 0 , , . . . , m −
1, let Γ (cid:48) i denote the rotated copy of Γ (cid:48) around the z -axis,with angle πim .Let P (cid:48) = conv (cid:83) m − i =0 Γ (cid:48) i . Then P (cid:48) is a convex polyhedral domainsuch that K \ P (cid:48) ⊆ K ( τ ), and if m is sufficiently large, then at any boundarypoint of P (cid:48) with z -coordinate greater than ρ − τ , | ( x, y, g ( x, y )) | is strictly locallyincreasing in a neighborhood of (0 ,
0) as a function of (cid:112) x + y , where bd( P (cid:48) ) isgiven as the graph of the function z = g ( x, y ). Thus, by Remark 4 and followingthe idea at the end of Subcase 1.2 in Step 2, we may truncate a neighborhood of q in bd( K ) by a convex polyhedral region P (cid:48) such that the only equilibrium pointof the truncated body on bd( P (cid:48) ) is the unstable point q (cid:48) , and the truncated bodyhas no non-truncated boundary point where some supporting plane contains anequilibrium point.The procedure discussed in Subcases 1.1-1.3 for q are applied for any equilibriumpoint in the orbit of q in an analogous way. Case 2 , if the stabilizer of q in G is not the identity. In this case the proceduredescribed in Case 1 is carried out in such a way that the truncating polyhedraldomain is invariant under any element of G fixing q .Summing up, to construct K in Step 2 we truncated a neighborhood of eachequilibrium point of K by a polyhedral region in such a way that each regioncontains exactly one equilibrium point relative to o , and no plane supporting K at any point of X = bd( K ) ∩ K contains an equilibrium point of K relative to o . In addition, K is G -invariant. Step 3 .In this step let Y = X ∩ X . Furthermore, for any plane H let o H denote theorthogonal projection of o onto H , and let H denote the family of planes H with ONOSTABLE POLYHEDRA 13 the property that o H ∈ K . Note that H consists of all planes through o , and forany p ∈ K \ { o } the (unique) plane passing through p and perpendicular to [ o, p ].Observe that Y is compact, and by our construction, for any plane H supporting K at some point p ∈ Y , we have o H / ∈ H ∩ K ; or equivalently, for any H ∈ H , H ∩ Y = ∅ . Thus, by compactness, there is some δ > H ∈ H and p ∈ Y , the distance of H and p is at least δ >
0. Now, for any p ∈ Y , let H p denote the unique closed half space whose boundary supports K at p and whichsatisfies int( H p ) ∩ K = ∅ . Let u p denote the outer unit normal vector of K at p ,and for any ζ >
0, set H p ( ζ ) = H p − ζu p , and Y ( ζ ) = K ∩ (cid:16)(cid:83) p ∈ Y H p ( ζ ) (cid:17) . Clearly, Y ( ζ ) tends to Y with respect to Hausdorff distance as ζ → + . Thus, there is somesufficiently small ζ > H ∈ H , H is disjoint from Y ( ζ ).Now, for any p ∈ Y , set U ( p ) = Y ∩ int( H p ( ζ )). Then U ( p ) is an open neigh-borhood of p in Y . Thus, by the compactness of Y , there are finitely many points p , . . . , p m such that (cid:83) mi =1 U ( p i ) = Y . Then, clearly P = K ∩ (cid:0)(cid:84) mi =1 (cid:0) R \ int( H p i ( ζ )) (cid:1)(cid:1) is a convex polytope contained in K . Furthermore, since ζ > P can be arbitrarily close to K .We show that no point q ∈ bd( P ), contained in some bd( H p i ( ζ )) is an equilib-rium point of P . Indeed, if q was such a point, then the plane H through q and per-pendicular to [ o, q ] is contained in H . On the other hand, q ∈ bd( P ) ∩ bd( H p i ( ζ )) ⊂ Y ( ζ ), which is impossible by our choice of ζ .Finally, we may choose the points p , p , . . . , p m in such a way that the set { p , . . . , p m } is invariant under the act of any element of G .5. Remarks and open questions
First, we remark that by using truncations instead of conic extensions in theproof of Theorem 1, we readily obtain Theorem 3. Here, a mono-unstable convexbody is meant to be a nondegenerate convex body with a unique unstable point.
Theorem 3.
For any n ≥ , n ∈ Z and ε > there is a homogeneous mono-unstable polyhedron P such that P has an n -fold rotational symmetry and d H ( P, B ) <ε . We ask the following.
Question 1.
What are the positive integers n ≥ , p containsa convex polyhedron with an n -fold axis of symmetry?In light of the words of Shephard in [28] from Section 1 about monostable poly-hedra, we remark that a consequence of Theorem 2 is that to study the metricproperties of nondegenerate polyhedra, in particular monostable polyhedra, it issufficient to study the metric properties of their smooth counterparts, which seemto be much more tractable.Next, we conjecture that Theorem 2 remains true if we omit the condition thatthe principal curvatures of bd( K ) at every equilibrium point of K are strictlypositive. Finally, to propose a conjecture for the first part of Problem 3, we recall thefollowing concept from [32], where the function ρ K : S → R , ρ K ( x ) = max { λ : λx ∈ K } is called the gauge function of the convex body K . Definition 1.
Let K ∈ R be a centered convex body, and for any simple, closedcurve Γ ⊂ S , let Γ + and Γ − denote the two compact, connected domains in S bounded by Γ. Then the quantities F ( K ) = sup Γ sup p ∈ Γ + ,p ∈ Γ − min { ρ K ( s ) : s ∈ Γ } max { ρ K ( p ) , ρ K ( p ) } and T ( K ) = sup Γ sup p ∈ Γ + ,p ∈ Γ − min { ρ K ( p ) , ρ K ( p ) } max { ρ K ( s ) : s ∈ Γ } are called the flatness and the thinness of K , respectively.Domokos and V´arkonyi in [32] proved that for any nondegenerate, centeredsmooth convex body K , F ( K ) = 1 if and only if K is monostable, and T ( K ) = 1if and only if K is mono-unstable.Recall that a nondegenerate convex body is mono-monostatic if it has a uniquestable and a unique unstable point [32]. We conjecture the following. Conjecture 1.
For any centered convex body K ⊂ R , K can be uniformly approx-imated by monostable convex polyhedra if and only if F ( K ) = 1, by mono-unstableconvex polyhedra if and only if T ( K ) = 1, and by mono-monostatic convex poly-hedra if and only if F ( K ) = T ( K ) = 1. References [1] A. Abramson et al.,
An ingestible self-orienting system for oral delivery of macromolecules ,Science (6427) (2019), 611–615.[2] I. B´ar´any, J. Itoh, C. Vˆılcu and T. Zamfirescu,
Every point is critical , Adv. Math. (2013),390-397.[3] A. Bezdek,
On stability of polyhedra , in: Workshop on Discrete Geometry, Sept. 13-16 2011,Fields Institute, Canada, 2490-2491.[4] K.F. Bohringer, B.R. Donald, L.E. Kavraki and F. Lamiraux,
Part orientation with oneor two stable equilibria using programmable vector fields , IEEE Transactions on RoboticsAutomation. (2) (2000), 157-170.[5] J.H. Conway and R.K. Guy, Stability of polyhedra , SIAM Rev. (3) (1966), 381.[6] H.T. Croft, K. Falconer and R.K. Guy, Unsolved Problems in Geometry , Springer, New York,NY, 1991.[7] R. Dawson,
Monostatic simplexes , Amer. Math. Monthly (1985), 541-546.[8] R. Dawson and W. Finbow, What shape is a loaded die? , Math. Intelligencer (1999),32-37.[9] R. Dawson and W. Finbow, Monostatic Simplexes III , Geom. Dedicata (2001), 101-113.[10] R. Dawson, W. Finbow and P. Mak, Monostatic simplexes II , Geom. Dedicata (1998),209-219.[11] G. Domokos, The G¨omb¨oc Pill , Math. Intelligencer (2019), 9–11.[12] G. Domokos, P. Holmes and Z. L´angi, A genealogy of convex solids via local and globalbifurcations of gradient vector fields , J. Nonlinear Sci. (2016), 1789-1815.[13] G. Domokos, F. Kov´acs, Z. L´angi, K. Reg˝os and P.T. Varga, Balancing polyhedra , Ars Math.Contemp., DOI:10.26493/1855-3974.2120.085[14] G. Domokos, Z. L´angi and T. Szab´o,
On the equilibria of finely discretized curves and sur-faces , Monatsh. Math. (2012) 321-345.[15] G. Domokos, Z. L´angi and T. Szab´o,
A topological classification of convex bodies , Geom.Dedicata (2016), 95-116.
ONOSTABLE POLYHEDRA 15 [16] G. Domokos, A. ´A. Sipos, T. Szab´o and P.L. V´arkonyi,
Pebbles, shapes and equilibria , Math.Geosciences (1) (2010), 29-47.[17] G. Domokos and P. V´arkonyi, Geometry and self-righting of turtles , Proc. R. Soc. B. (2007), 11-17.[18] A. Dumitrescu and C.D. T´oth,
On the cover of the rolling stone , Proceedings of the 31stAnnual ACM-SIAM Symposium on Discrete Algorithms (2020), 2575-2586.[19] F. Galaz-Garc´ıa and L. Guijarro,
Every point in a Riemannian manifold is critical , Calc.Var. (2015), 2079–2084.[20] M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy) , SIAM Rev. The Works of Archimedes , Cambridge University Press, 1897.[23] A. Heppes,
A double-tipping tetrahedron , SIAM Rev. (1967), 599-600.[24] J. Itoh, C. Vˆılcu and T. Zamfirescu, With respect to whom are you critical? , Adv. Math. (2020), 107187.[25] M.S. Klamkin,
Problems in Applied Mathematics , SIAM, Philadelphia, USA, 1990.[26] A. Reshetov,
A unistable polyhedron with faces (English summary) , Internat. J. Comput.Geom. Appl. (2014), 3959.[27] R. Schneider, Convex bodies: the Brunn-Minkowski theory , Encyclopedia of Mathematicsand its Applications , Cambridge University Press, Cambridge, 1993.[28] G.C. Shephard, Twenty problems on convex polyhedra part II , Math. Gaz. (1968), 359-367.[29] T. Szab´o and G. Domokos, A new classification system for pebble and crystal shapes basedon static equilibrium points , Central European Geology, (1) (2010), 1-19.[30] T. Szab´o, G. Domokos, J.P. Grotzinger and D.J. Jerolmack, Reconstructing the transporthistory of pebbles on Mars , Nature Comm. (2015), paper 8366.[31] P.L. V´arkonyi, Estimating Part Pose Statistics With Application to Industrial Parts Feedingand Shape Design: New Metrics, Algorithms, Simulation Experiments and Datasets , IEEETransactions on Automation Science and Engineering (3) (2014), 658 - 667.[32] P.L. V´arkonyi and G. Domokos, Static equilibria of rigid bodies: dice, pebbles and thePoincar´e-Hopf Theorem , J. Nonlinear Sci. (2006), 255-281.[33] T. Zamfirescu, How do convex bodies sit? , Mathematika (1995), 179-181.[34] A. Zomorodian, Topology for computing , Cambridge University Press, 2005.
MTA-BME Morphodynamics Research Group and Department of Geometry, BudapestUniversity of Technology, Egry J´ozsef utca 1., Budapest 1111, Hungary
Email address ::