Isoperimetric problems for three-dimensional parallelohedra and translative, convex mosaics
IISOPERIMETRIC PROBLEMS FOR THREE-DIMENSIONALPARALLELOHEDRA AND TRANSLATIVE, CONVEX MOSAICS
ZSOLT L ´ANGI
Abstract.
The aim of this note is to investigate isoperimetric-type problemsfor 3-dimensional parallelohedra; that is, for convex polyhedra whose translatestile the 3-dimensional Euclidean space. Our main result states that among 3-dimensional parallelohedra with unit volume the one with minimal mean widthis the regular truncated octahedron. In addition, we establish a connectionbetween the edge lengths of 3-dimensional parallelohedra and the edge densitiesof the translative mosaics generated by them, and use our method to provethat among translative, convex mosaics generated by a parallelohedron witha given volume, the one with minimal edge density is the face-to-face mosaicgenerated by cubes. Introduction
Among the convex polyhedra in Euclidean 3-space R most known both insideand outside mathematics, we find the so-called parallelohedra ; that is, the convexpolyhedra whose translates tile space. All parallelohedra in 3-space can be definedas the Minkowski sums of at most six segments with prescribed linear dependen-cies between the generating segments, and thus, they are a subclass of zonotopes .Parallelohedra in R are also related to the Voronoi cells of lattices via Voronoi’sconjecture, stating (and proved in R , see [6, 10, 12]) that any parallelohedronis an affine image of such a cell. Well-known examples of parallelohedra are thecube, the regular rhombic dodecahedron, and the regular truncated octahedron,which are the Voronoi cells generated by the integer, the face-centered cubic andthe body-centered cubic lattice, respectively.Many papers investigate isoperimetric problems for zonotopes (see, e.g. [1, 2,21]), with special attention on the relation between mean width and volume (see[5, 9]). On the other hand, apart from the celebrated proof of Kepler’s Conjectureby Hales [16], which, as a byproduct, implies that among parallelohedra with a giveninradius, the one with minimum volume is the regular rhombic dodecahedron, thereis no known isoperimetric-type result for parallelohedra.In our paper we offer a new representation of 3-dimensional parallelohedra thatmight be useful to investigate their geometric properties. We use this representationto prove Theorem 1. Mathematics Subject Classification.
Key words and phrases. parallelohedra, zonotopes, discrete isoperimetric problems, density,tiling, Kepler’s conjecture, honeycomb conjecture, Kelvin’s conjecture.The author is supported by the National Research, Development and Innovation Office, NKFI,K-134199, the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, andgrants BME IE-VIZ TKP2020 and ´UNKP-20-5 New National Excellence Program by the Ministryof Innovation and Technology. a r X i v : . [ m a t h . M G ] F e b Z. L´ANGI
Theorem 1.
Among unit volume -dimensional parallelohedra, regular truncatedoctahedra have minimal mean width. By their definition, parallelohedra are closely related to translative, convex mo-saics in R [29], which are the topic of the second part of the paper. During ourinvestigation, we denote the family of translative, convex mosaics whose cells haveunit volume by M . To state our results, we define the edge density and the surfacedensity of a mosaic in the usual way: if M is a mosaic and i ∈ { , } , and the limit ρ i ( M ) = lim sup r →∞ vol i (skel i ( M ) ∩ ( r B ))vol ( r B )exists, where B is the Euclidean unit ball centered at the origin o , vol i ( · ) denotes i -dimensional volume, and skel i ( M ) denotes the i -dimensional skeleton of M , then ρ ( M ) is called the (upper) edge density , and ρ ( M ) is called the (upper) sur-face density of M . As a special case of the relation between parallelohedra andtranslative, convex mosaics, one may expect that for a parallelohedron P and theassociated face-to-face, translative, convex mosaic M , the edge density of M andthe mean width of P , and similarly the surface density of M and the surface of P ,are related in some way. In the second part of the paper we explore this connection,and, using a suitable modification of the proof of Theorem 1, we prove Theorem 2. Theorem 2.
A translative, convex mosaic M ∈ M has minimal edge density ifand only if M is a face-to-face mosaic with cubes as cells. The structure of the paper is as follows. In Section 2 we introduce and pa-rametrize 3-dimensional parallelohedra, and describe the tools and the notationnecessary to prove our results. In Section 3 we prove Theorem 1. In Section 4 weinvestigate the properties of translative, convex mosaics, and show how to modifythe arguments in Section 3 to prove Theorem 2. In Section 5, we collect additionalremarks, and recall some old, and raise some new problems. Finally, in our Ap-pendix we collect the results of our computations for the minima of the quantitiesinvestigated in this paper for the different types of parallelohedra and translative,convex mosaics. 2.
Preliminaries
Properties of three-dimensional parallelohedra.
Minkowski [24] and Ven-kov [33] proved that P is a convex n -dimensional polytope for which there is aface-to-face tiling of R n with translates of P if and only if(i) P and all its facets are centrally symmetric, and(ii) the projection of P along any of its ( n − P . The polytopessatisfying properties (i-ii) are called n -dimensional parallelohedra .The combinatorial classes of 3-dimensional parallelohedra are well known (cf.Figure 1). More specifically, any 3-dimensional parallelohedron is combinatoriallyisomorphic to one of the following:(1) a cube,(2) a hexagonal prism, SOPERIMETRIC PROBLEMS 3 (3) Kepler’s rhombic dodecahedron (which we call a regular rhombic dodeca-hedron),(4) an elongated rhombic dodecahedron,(5) a regular truncated octahedron.We call a parallelohedron combinatorially isomorphic to the polyhedron in (i) a type (i) parallelohedron . It is also known that every parallelohedron P in R is azonotope. More specifically, a type (1)-(5) parallelohedron can be attained as theMinkowski sum of 3 , , , , P ⊂ R is a zonotope, every edge of P is a translate of one of the segments generating P . Furthermore, for any generatingsegment S , the faces of P that contain a translate of S form a zone , i.e. they canbe arranged in a sequence F , F , . . . , F k , F k +1 = F of faces of P such that for allvalues of i , F i ∩ F i +1 is a translate of S . By property (ii) in the list in the beginningof Section 2, this sequence contains 4 or 6 faces. In these cases we say that S generates a 4-belt or a 6-belt in P , respectively. The numbers of 4- and 6-belts ofa type (i) parallelohedron are 3 and 0, 3 and 1, 0 and 4, 1 and 4, and 0 and 6 for i = 1 , , , ,
5, respectively.
Type (2)
Type (1)
Type (3)
Type (4)
Type (5)
Figure 1.
The five combinatorial types of 3-dimensional paral-lelohedra. The type (5) parallelohedron in the picture is the regu-lar truncated octahedron generated by the six segments connectingthe midpoints of opposite edges of a cube. The type (3) polyhe-dron is the regular rhombic dodecahedron generated by the fourdiagonals of the same cube. The rest of the polyhedra in the pic-ture are obtained by removing some generating segments from thetype (5) polyhedron.
Z. L´ANGI
Remark 1.
Let P = (cid:80) mi =1 [ o, v i ] ⊂ R be a parallelohedron, and set I = { , , . . . , m } .Then the volume, the surface area and the mean width of P is(1) vol ( P ) = (cid:88) { i,j,k }⊂ I | V ijk | , (2) surf( P ) = 2 (cid:88) { i,j }⊂ I | v i × v j | , and(3) w ( P ) = m (cid:88) i =1 | v i | , respectively, where V ijk is the determinant of the matrix with column vectors v i , v j , v k . Proof.
The formula in (1) is the well-known volume formula for zonotopes (cf. e.g.[31]). The one in (2) can be proved directly for type (5) parallelohedra, whichimplies its validity for all parallelohedra. Finally, Steiner’s formula [28] yields thatthe second quermassintegral W ( P ) of P is (cid:80) ij ( π − α ij ) l ij , where α ij and l ij isthe dihedral angle and the length of the edge between the i th and j th faces of P .Thus, from the zone property it follows that W ( P ) = π (cid:80) mi =1 | v i | . On the otherhand, for any convex body K ⊂ R we have w ( K ) = π W ( K ), which implies(3). (cid:3) If we choose the generating segments in the form [ o, p i ] for some vectors p i ∈ R , then in case of a type (1) or type (4) parallelohedron, the vectors v i are ingeneral position, i.e. any three of them are linearly independent, whereas in caseof the other types there are triples of vectors that are restricted to be co-planar.In particular, if P is a type (5) parallelohedron, then each of the four pairs ofhexagon faces defines a linear dependence relation on the generating segments.More specifically, if v , v , v , v are normal vectors of the four pairs of hexagonfaces of the parallelohedron, then the directions of the six generating segments arethe ones defined by the six cross-products v i × v j , i (cid:54) = j . Note that all triples ofthe v i s are linearly independent, as otherwise some of the generating segments of P are collinear. Thus, we have(4) P = (cid:88) ≤ i
It is easy to check that in the remaining cases P is planar.Thus, we can use this representation for all parallelohedra appearing in the paperwith the assumption that β ij ≥ ≤ i < j ≤ v i are linearly independent, but clearly all fourof them are, up to multiplying by a constant they have a unique nontrivial linearcombination λ v i + λ v + λ v + λ v = o . Since in the representation of P weconsidered only the directions of the v i s, we may clearly assume that v + v + v + v = o . This implies that the tetrahedron conv { v , v , v , v } is centered , that is,its center of gravity is o . From the equation v + v + v + v = o it follows that theabsolute values of the determinants of any three of the v i s are equal. We choose thiscommon value 1, which yields that the volume of the tetrahedron is . Throughoutthe paper for any i, j, k ∈ { , , , } , we denote by V ijk the value of the determinantwith v i , v j , v k as column vectors. We choose the indices in such a way that V = 1.Since v + v + v + v = o , we have that for any { i, j, s, t } = { , , , } , the planecontaining o, v i , v j strictly separates v s and v t , which implies that V ijs = − V ijt .In the proof we often use the function f : R → R ,(5) f ( τ , τ , τ , τ , τ , τ ) = τ τ τ + τ τ τ + τ τ τ + τ τ τ ++( τ + τ )( τ τ + τ τ )+( τ + τ )( τ τ + τ τ )+( τ + τ )( τ τ + τ τ ) . We remark that f is not the third elementary symmetric function on six variables,since after expansion it has only 16 <
20 = (cid:0) (cid:1) members.An elementary computation yields that for any i, j, k, l, s, t ∈ { , , , } , we have | v i × v j , v k × v l , v s × v t | = V ijl V stk − V ijk V stl . Using this formula and the properties of cross-products, we may express the volume,surface area and mean width of P in our representation. Remark 2.
For the parallelohedron P = (cid:80) i (cid:54) = j β ij [ o, v i × v j ] satisfying the condi-tions above, we have(6) vol ( P ) = f ( β , β , β , β , β , β ) , (7) surf( P ) = 2 (cid:0) ( β β + β β + β β ) | v | + . . . ( β β + β β + β β ) | v | ++ β β ( | v + v | ) + . . . + β β | v + v | (cid:1) , (8) w ( P ) = (cid:88) ≤ i The following lemma is used more than once in thepaper. Lemma 1. Let T = conv { p , p , p , p } be an arbitrary centered tetrahedron withvolume V > where the vertices are labelled in such a way that the determinant withcolumns p , p , p is positive. For any { i, j, s, t } = { , , , } , let γ ij = −(cid:104) p s , p t (cid:105) and ζ ij = γ ij | p i × p j | . Then (1) f ( γ , γ , γ , γ , γ , γ ) = V , (2) (cid:80) ≤ i Proof. Let χ ij = (cid:104) p i , p j (cid:105) for all i, j . Consider the Gram matrix G defined by p , p , p , and observe that as T is centered, the volume of the parallelepipedspanned by them is V . Since the determinant of the Gram matrix of d linearlyindependent vectors in R n is the square of the volume of the parallelotope spannedby the vectors, it follows that det( G ) = V . Furthermore, since T is centered,we have χ = − χ − χ − χ , and we may obtain similar formulas for χ and χ . Now, substituting these formulas into f ( γ , . . . , γ ) = f ( − χ , . . . , − χ ),we obtain that f ( γ , . . . , γ ) = χ χ χ + 2 χ χ χ − χ χ − χ χ − χ χ = det( G ) , which implies the first identity.To obtain the second identity, observe that for any { i, j, s, t } = { , , , } , wehave ζ ij = −(cid:104) p s , p t (cid:105)| p i × p j | = − χ st (cid:0) χ ii χ jj − χ ij (cid:1) . This observation and anargument similar to the one in the previous paragraph yields that (cid:80) ≤ i Under the condition that (cid:80) ≤ i Recall from Subsection 2.1 that we represent P in the form P = (cid:88) ≤ i 0. By ourassumptions, | V ijk | = 1 for any { i, j, k } ⊂ { , , , } , where V ijk denotes the deter-minant with v i , v j , v k as columns, and we assumed that V = 1, which implies, inparticular, that V = − 1. Then T = conv { v , v , v , v } is a centered tetrahedron,with volume vol ( T ) = .By Remark 2, we need to find the minimum value of w ( P ) = (cid:80) ≤ i Z. L´ANGI Lemma 3. Let P ⊂ R n be a convex polytope with outer unit facet normals u , . . . , u k .Let F i denote the ( n − -dimensional volume of the i th facet of P . Then, up tocongruence, there is a unique affine transformation L such that surf( L ( P )) is max-imal in the affine class of P . Furthermore, P satisfies this property if and only ifits surface area measure is isotropic, that is, if (9) k (cid:88) i =1 nF i surf( P ) u i ⊗ u i = Id where Id denotes the identity matrix. Any convex polytope satisfying the conditions in (9) is said to be in surfaceisotropic position . We note that the volume of the projection body of any convexpolyhedron is invariant under volume preserving linear transformations (cf. [27]).On the other hand, from Cauchy’s projection formula and the additivity of thesupport function (see. [13]) it follows that the projection body of the polytopein Lemma 3 is the zonotope (cid:80) ki [ o, F i u i ]. Note that the solution to Minkowski’sproblem [28] states that some unit vectors u , . . . , u k ∈ R n and positive numbers F , . . . , F k are the outer unit normals and volumes of the facets of a convex poly-tope if and only if the u i s span R n , and (cid:80) ki =1 F i u i = o . On the other hand, atranslate of the parallelohedron P in our investigation can be written in the form (cid:80) ≤ i 4. Since u ⊗ u = ( − u ) ⊗ ( − u ) for any u ∈ R n , any solution P to our optimization prob-lem satisfies the conditions in (9) with the vectors v i × v j | v i × v j | in place of the u i s, andthe quantities β ij | v i × v j | in place of the F i s. Thus, applying also the formula inRemark 2 for w ( P ), it follows that(10) (cid:88) ≤ i 27 vol ( P )( w ( P )) = f ( ζ , . . . , ζ ), where ζ ij is defined as above,in the family of all centered tetrahedra T with vol ( T ) = , under the conditionthat (cid:80) ≤ i Step 2 .Consider the function f ( ζ , . . . , ζ ) , where ζ ij = −(cid:104) v s , v t (cid:105)| v i × v j | for { i, j, s, t } = { , , , } , and v , v , v , v are the vertices of a centered tetrahedron T with volume . Set γ ij = −(cid:104) v s , v t (cid:105) and τ ij = γ ij | v i × v j | for all { i, j, s, t } = { , , , } . To givean upper bound on the value of f , we apply the Cauchy-Schwartz Inequality,which states that for any nonnegative real numbers x i , y i , i = 1 , , . . . , k , wehave (cid:80) ki =1 x i y i ≤ (cid:113)(cid:80) ki =1 x i (cid:113)(cid:80) ki =1 y i , with equality if and only if the x i s andthe y i s are proportional. To do this, we write each member of f as the product ζ ij ζ kl ζ mn = √ γ ij γ kl γ mn √ τ ij τ kl τ mn . Thus, we obtain that f ( ζ , . . . , ζ ) ≤ (cid:112) f ( γ , . . . , γ ) (cid:112) f ( τ , . . . , τ ) = (cid:112) f ( τ , . . . , τ ) , where we used the fact that by Lemma 1, f ( γ , . . . , γ ) = 1 for all centered tetrahe-dra with volume . Furthermore, observe that by Lemma 1 we have (cid:80) ≤ i ≤ j ≤ τ ij =3 for all such tetrahedra. Hence, by Lemma 2, we have f ( ζ , . . . , ζ ) ≤ √ f ( ζ , . . . , ζ ) = √ 2. Then, by Lemma 2, τ ij = for allvalues i (cid:54) = j . This, by the Cauchy-Schwartz Inequality, implies that for some t ∈ R , γ ij = t for all values i (cid:54) = j . Since T is centered, this implies that γ ii = 3 t for all i s.In other words, the Gram matrix of the vectors v , . . . , v is a scalar multiple of thematrix 4 Id − E , where E is the matrix with all entries equal to 1. Since the Grammatrix of a vector system determines the vectors up to orthogonal transformations,and it is easy to check that 4 Id − E is the Gram matrix of the vertex set of acentered regular tetrahedron of circumradius √ 3, the equality part in Theorem 1follows.4. Translative, convex mosaics and the proof of Theorem 2 Our next lemma establishes a relationship between the edge density of a transla-tive, convex mosaic M and the edge lengths of the generating parallelohedron P . Lemma 4. Let P = (cid:80) ki =1 [ o, p i ] be a unit volume parallelohedron in R , and let M be a tiling of R with translates of P . Set w i = (cid:26) , if the projection of P along p i is a parallelogram, , if the projection of P along p i is a centrally symmetric hexagon.Then (11) ρ ( M ) ≥ (cid:88) i =1 w i | p i | with equality if M is face-to-face.Proof. First, we assume that M is face-to-face and P is a cell of M , and choose anedge E of P . Let E contain the edges E (cid:48) of the tiling for which there is a sequenceof cells P , . . . , P k in which consecutive members have a common face and this facecontains an edge parallel to E , and E (cid:48) ⊂ P and E (cid:48) ⊂ P k . Furthermore, let P bethe family of cells containing edges from E . Then the projection of the members of P in the direction of E onto a plane is a translative, convex, edge-to-edge tiling ofthe plane. By the second property of parallelohedra in the list in the beginning ofSection 2, the projections of the elements of P are either parallelograms, in which every vertex belongs to exactly four parallelograms, or they are centrally symmetrichexagons, in which every vertex belongs to three hexagons. Thus, a cell contains 4or 6 translates of any given edge, and in the first case every edge belongs to exactly4 cells, and in the second case every edge belongs to exactly 3 cells.Now, we assign a weight w E to each edge E of M ; we set w ( E ) = k , where k is thenumber of cells E belongs to. Let N r denote the number of cells of M , and E r denotethe family of the edges of the cells contained in r B . As N r = vol ( r B ) + O ( r ),we havevol (skel ( M ) ∩ ( r B )) = (cid:88) E ∈E r vol ( E ) + O ( r ) = N r (cid:88) i =1 w i | p i | + O ( r ) . Finally, if M (cid:48) is a not necessarily face-to-face tiling by translates of P , then we maydivide each edge into finitely many pieces in such a way that relative interior pointson each piece belong to the same cells. It is easy to see that any part of an edge in a6-belt belongs to exactly 3 cells, and a part of an edge in a 4-belt belongs to either4 or 2 cells. Thus, repeating the procedure, the weighted sum of the edge lengthsin r B , defined like in the previous case, yields a value for ρ ( M ) asymptoticallynot less than that for a face-to-face mosaic generated by P . (cid:3) Now we prove Theorem 2. First, we consider only face-to-face mosaics. Let P be the parallelohedron generating M . It is easy to see that if P is a type (1)parallelohedron, then ρ ( M ) is minimal if and only if P is a cube, and in thiscase ρ ( P ) = 3. Consider the case that P is a type (2) parallelohedron; that is, acentrally symmetric hexagon based prism. If the lengths of the edges of the hexagonare a , a , a , and the length of its lateral edges is b , then ρ ( M ) = a + a + a + 2 b .An elementary computation shows that if vol ( P ) = 1, then this quantity is minimalif and only if P is a right prism, its base is a regular hexagon and 4 b = 3 a = 3 a =3 a . Furthermore, in this case ρ ( M ) = 3 / ≈ . 60. Next, consider the case that P is a type (3) or a type (5) parallelohedron. Then all edges of P belong to 6-belts,and hence, ρ ( M ) = 2 w ( P ), where w ( P ) denotes the mean width of P . Thus, byTheorem 1, we have that in this case ρ ( M ) ≥ / · ≈ . P in the form P = (cid:88) ≤ i 0, withexactly one β ij equal to zero. By our assumptions, | V ijk | = 1 for any { i, j, k } ⊂{ , , , } , where V ijk denotes the determinant with v i , v j , v k as columns, and weassumed that V = 1, which implies, in particular, that V = − 1. Then T =conv { v , v , v , v } is a centered tetrahedron, with volume vol ( T ) = .Without loss of generality, we set β = 0. Then, among the edges of P , thetranslates of β v × v belong to a 4-belt, and the translates of all other edgesbelong to 6-belts. Thus, we have ρ ( M ) = β | v × v | + 2 (cid:88) i ∈{ , } ,j ∈{ , } β ij | v i × v j | , andvol ( P ) = f ( β , β , . . . , β , . SOPERIMETRIC PROBLEMS 11 We intend to find the minimum of ρ ( M ) under the condition that vol ( P ) = 1.We carry out the same two steps as in the proof of Theorem 1. In particular, weobserve that there is an o -symmetric convex polytope in R with outer unit facetnormals ± v i × v j | v i × v j | where i (cid:54) = j and { i, j } (cid:54) = { , } , and areas of the correspondingfacets β | v × v | , β | v × v | , . . . , β | v × v | . Similarly like in the previoussection, we may assume that this polytope is in surface isotropic position, implyingthat β | v × v | ( v × v ) ⊗ ( v × v )+ (cid:88) i ∈{ , } ,j ∈{ , } β ij | v i × v j | ( v i × v j ) ⊗ ( v i × v j ) = ρ ( M )3 Id . Multiplying this equation both from the left and from the right by some of the v i s,we obtain that β = −(cid:104) v , v (cid:105)| v × v | · ρ ( M )3 , and for any i ∈ { , } and j ∈ { , } , we have β ij = −(cid:104) v , v (cid:105)| v × v | · ρ ( M )6 . Let ζ ij = −(cid:104) v s , v t (cid:105)| v i × v j | be defined as in the previous section. Then our opti-mization problem can be rewritten as finding the maximum of216 vol ( P )( ρ ( M )) = f (2 ζ , ζ , . . . , ζ , T with vol ( T ) = and (cid:104) v , v (cid:105) = 0, underthe condition that (cid:80) ≤ i 0) = 1.Then a calculation similar to that in Lemma 2 shows that for any values τ ij ≥ τ = 0 and (cid:80) ij τ ij = 3, we have f (4 τ , τ , τ , τ , τ , ≤ , from which ρ ( M ) ≥ / · / / ≈ . 66 if M is generated by a type (4) parallelohe-dron. This finishes the proof for face-to-face mosaics.Now, we have seen in the proof of Lemma 4 that if M and M (cid:48) are translative,convex mosaics in R generated by a paralelohedron P , and M is face-to-face, then ρ ( M ) ≤ ρ ( M (cid:48) ). On the other hand, it is also easy to see that if P is a cubeand M (cid:48) is not face-to-face, then ρ ( M ) < ρ ( M (cid:48) ). This yields the equality part ofTheorem 2. 5. Remarks and open problems For a convex polyhedron P ⊂ R , let E ( P ) denote the total edge length of P .Melzak [22] in 1965 made the conjecture that for any unit volume convex polyhedron P ⊂ R , the total edge length of P is E ( P ) ≥ / · / ≈ . 89, with equality for certain regular triangle based prisms. This conjecture is still open despite beingthe subject of scientific research even now [30].A slight modification of the proof of Theorem 2 provides a proof of this conjecturefor the special case of parallelohedra. Remark 3. For any unit volume parallelohedron P in R , we have E ( P ) ≥ , with equality if and only if P is a cube.The origin of the Honeycomb Conjecture, stating that in a decomposition of theEuclidean plane into regions of equal area, the regular hexagonal grid has the leastperimeter, can be traced back to ancient times [32]. This problem has been in thefocus of research throughout the second half of the 20th century [8, 23], and wasfinally settled by Hales [15].The most famous analogous conjecture for mosaics in 3-dimensional Euclideanspace is due to Lord Kelvin [19], who in 1887 conjectured that in a tiling of spacewith cells of unit volume, the mosaic with minimal surface area is composed ofslightly modified truncated octahedra. Even though this conjecture was disprovedby Weaire and Phelan in 1994 [34], who discovered a tiling of space with two slighlycurved ‘polyhedra’ of equal volume and with less total surface area than in LordKelvin’s mosaic, the original problem of finding the mosaics with equal volume cellsand minimal surface area has been extensively studied (see, e.g. [20, 11, 25]). On theother hand, in the author’s knowledge, there is no subfamily of mosaics for whichKelvin’s problem is solved. This is our motivation for the following conjecture.Before we state it, we observe that for any (not necessarily face-to-face) translative,convex mosaic M of R , generated by translates of a parallelohedron P , we have ρ ( M ) = surf( P ). Conjecture 1. Among translative, convex mosaics M in R generated by a unitvolume parallelohedron P , ρ ( M ) is minimal if and only if P is a regular truncatedoctahedron.The following conjecture, called Rhombic Dodecahedral Conjecture, can be foundin [1]. We note that this conjecture is the lattice variant of the so-called StrongDodecahedral Conjecture [3] proved by Hales in [17]. For other variants of theDodecahedral Conjecture, see also [4] or [18]. Conjecture 2 (Bezdek, 2000) . The surface area of any parallelohedron in R withunit inradius is at least as large as 12 √ ≈ . 97, which is the surface area of theregular rhombic dodecahedron of unit inradius.As it was mentioned in the introduction, Voronoi cells of lattice packings of con-gruent balls is an important subfamily of 3-dimensional parallelohedra. Regardingthis subclass, Dar´oczy [5] gave an example of a packing whose Voronoi cells havea smaller mean width than that of the regular rhombic dodecahedron of the sameinradius. This is the motivation behind our last question. Here we note that themean widths of a cube, a regular truncated octahedron and a regular rhombic do-decahedron of unit inradius is 6 > √ √ 6, respectively, which in our opinion,makes the question indeed intriguing. Problem 1. Find the minimum of the mean widths of 3-dimensional parallelohedraof unit inradius. SOPERIMETRIC PROBLEMS 13 Appendix In three tables we collected the results of our investigation for the minimal valuesof the mean width and the total edge length of the different types of parallelohedra,and the edge density of translative, convex mosaics defined by different types ofparallelohedra.Recall (cf. Section 2) that the different types of parallelohedra correspond to thefollowing polyhedra. • Type (1) parallelohedra: parallelopipeds. • Type (2) parallelohedra: hexagonal prisms. • Type (3) parallelohedra: rhombic dodecahedra. • Type (4) parallelohedra: elongated rhombic dodecahedra. • Type (5) parallelohedra: truncated octahedra.Finally, note that while the mean width w ( · ) is a continuous function of itsvariable with respect to Hausdorff distance, the quantities ρ ( · ) and E ( · ) are not.Type Minimum of w ( P ) Optimal parallelohedra(1) 3 cube(2) / / ≈ . 86 regular hexagon based right prism, withbase and lateral edges of lengths / / and / / , respectively(3) 2 / · / ≈ . 75 regular rhombic dodecahedron with edgelength √ / (4) ≥ / / ≈ . 73 not known(5) / ≈ . 67 regular truncated octahedron of edgelength / Table 1. Minima of the mean widths of different types of paral-lelohedraType Infimum of ρ L ( P ) Optimal parallelohedra(1) 3 cube(2) 3 / ≈ . 60 regular hexagon based right prism, withbase and lateral edges of lengths / and / , respectively(3) 2 / · / ≈ . 50 regular rhombic dodecahedron with edgelength √ / (4) ≥ / · / / ≈ . 66 not known(5) 2 / · ≈ . 35 regular truncated octahedron with edgelength / . Table 2. Infima of the edge densities of translative, convex mo-saics generated by different types of parallelohedra Type Infimum of E ( P ) Optimal parallelohedra(1) 12 cube(2) 3 / / ≈ . 09 regular hexagon based right prism, withbase and lateral edges of length √ √ (3) 2 / · / ≈ . 50 regular rhombic dodecahedron with edgelengths √ / (4) ≥ / ≈ . 70 not known(5) 2 · ≈ . 04 regular truncated octahedron with edgelength / Table 3. Infima of the total edge lengths of different types ofparallelohedra Acknowledgements The author is indebted to A. Jo´os and G. 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