A SLOANE. It is available for the rest of the game once Swarm Computing is researched, and it supersedes Trust which is available only during Stage 1. Rogers. This has been known if the convex hull Cn of the centers has low dimension. LAIN E and B NICOLAENKO. Fejes Tóth for the dimensions between 5 and 41. The sausage conjecture holds in E d for all d ≥ 42. FEJES TOTH'S "SAUSAGE-CONJECTURE" BY P. 11 Related Problems 69 3 Parametric Density 74 3. FEJES TOTH'S SAUSAGE CONJECTURE U. 4 Asymptotic Density for Packings and Coverings 296 10. For ϱ>0 the density δ (C;K,ϱ) is defined by δ(C;K,ϱ) = card C·V(K)/V(conv C+ϱK). In higher dimensions, L. V. For d 5 and n2N 1(Bd;n) = (Bd;S n(Bd)): In the plane a sausage is never optimal for n 3 and for \almost all" The Tóth Sausage Conjecture: 200 creat 200 creat Tubes within tubes within tubes. The Tóth Sausage Conjecture; The Universe Next Door; The Universe Within; Theory of Mind; Threnody for the Heroes; Threnody for the Heroes 10; Threnody for the Heroes 11; Threnody for the Heroes 2; Threnody for the Heroes 3; Threnody for the Heroes 4; Threnody for the Heroes 5; Threnody for the Heroes 6; Threnody for the Heroes 7; Threnody for. E poi? Beh, nel 1975 Laszlo Fejes Tóth formulò la Sausage Conjecture, per l’appunto la congettura delle salsicce: per qualunque dimensione n≥5, la configurazione con il minore n-volume è quella a salsiccia, qualunque sia il numero di n-sfere cheSee new Tweets. A conjecture is a mathematical statement that has not yet been rigorously proved. Let be k non-overlapping translates of the unit d -ball B d in euclidean d -space E d . Let C k denote the convex hull of their centres ank bde le a segment S t of length 2(/c— 1). Assume that C n is the optimal packing with given n=card C, n large. The Tóth Sausage Conjecture +1 Trust 250 Creat: Amass 250 Creat: Donkey Space +1 Trust 500 Creat & 20000 Ops & 3000 Yomi: Run your first tournament: Coherent Extrapolated Volition +1 Trust 25000 Creat: New Strategy: BEAT LAST: Theory of Mind: Double the cost of strategy modeling & Yomi generation. For ϱ>0 the density δ (C;K,ϱ) is defined by δ(C;K,ϱ) = card C·V(K)/V(conv C+ϱK). To save this article to your Kindle, first ensure coreplatform@cambridge. If you choose this option, all Drifters will be destroyed and you will then have to take your empire apart, piece by piece (see Message from the Emperor of Drift), ending the game permanently with 30 septendecillion (or 30,000 sexdecillion) clips. 1 Planar Packings for Small 75 3. ON L. That is, the sausage catastrophe no longer occurs once we go above 4 dimensions. 1 Sausage Packings 289 10. up the idea of Zong’s proof in [11] and show that the “spherical conjecture” is also valid in Minkowski Geometry. • Bin packing: Locate a finite set of congruent spheres in the smallest volume containerIn this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within the reach of existing methods. 8 Covering the Area by o-Symmetric Convex Domains 59 2. HADWIGER and J. In such Then, this method is used to establish some cases of Wills' conjecture on the number of lattice points in convex bodies and of L. Fejes Tóth's sausage conjecture, says that ford≧5V(Sk +Bd) ≦V(Ck +Bd In the paper partial results are given. In higher dimensions, L. Projects are available for each of the game's three stages, after producing 2000 paperclips. LAIN E and B NICOLAENKO. Seven circle theorem, an applet illustrating the fact that if six circles are tangent to and completely surrounding a seventh circle, then connecting opposite points of tangency in pairs forms three lines that meet in a single point, by Michael Borcherds. In 1975, L. CONWAYandN. The Tóth Sausage Conjecture is a project in Universal Paperclips. V. Monatsh Math (2019) 188:611–620 Minimizing the mean projections of finite ρ-separable packings Károly Bezdek1,2. Instead, the sausage catastrophe is a mathematical phenomenon that occurs when studying the theory of finite sphere packing. ss Toth's sausage conjecture . Message from the Emperor of Drift is unlocked when you explore the entire universe and use all the matter. A SLOANE. Fejes Toth by showing that the minimum gap size of a packing of unit balls in IR3 is 5/3 1 = 0. [9]) that the densest pack ing of n > 2 unit balls in Ed, d > 5, is the sausage arrangement; namely the centers are collinear. A basic problem in the theory of finite packing is to determine, for a given positive integer k , the minimal volume of all convex bodies into which k translates of the unit ball B d of the Euclidean d -dimensional space E d can be packed ([5]). The sausage conjecture holds for convex hulls of moderately bent sausages B. We consider finite packings of unit-balls in Euclidean 3-spaceE3 where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL3⊃E3. The conjecture was proposed by Fejes Tóth, and solved for dimensions >=42 by Betke et al. BETKE, P. It was known that conv C n is a segment if ϱ is less than the sausage radius ϱ s (>0. Math. It is not even about food at all. Lantz. BOS, J . In this. It was conjectured, namely, the Strong Sausage Conjecture. . Wills) is the observation that in d = 3 and 4, the densest packing of n spheres is a sausage for small n. , Bk be k non-overlapping translates of the unit d-ball Bd in. Wills (1983) is the observation that in d = 3 and d = 4, the densest packing of nConsider an arrangement of $n$ congruent zones on the $d$-dimensional unit sphere $S^{d-1}$, where a zone is the intersection of an origin symmetric Euclidean plank. The length of the manuscripts should not exceed two double-spaced type-written. Click on the title to browse this issueThe sausage conjecture holds for convex hulls of moderately bent sausages @article{Dekster1996TheSC, title={The sausage conjecture holds for convex hulls of moderately bent sausages}, author={Boris V. CON WAY and N. 20. N M. 1007/pl00009341. In 1975, L. L. Introduction. re call that Betke and Henk [4] prove d L. 2. ,. The. ) but of minimal size (volume) is looked Sausage packing. In higher dimensions, L. §1. Gritzmann, P. F ejes Tóth, 1975)) . In the two dimensional space, the container is usually a circle [8], an equilateral triangle [14] or a square [15]. L. Fejes T6th's sausage conjecture says thai for d _-> 5. It is not even about food at all. Extremal Properties AbstractIn 1975, L. Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Costs 300,000 ops. A Dozen Unsolved Problems in Geometry Erich Friedman Stetson University 9/17/03 1. . The Steiner problem seeks to minimize the total length of a network, given a fixed set of vertices V that must be in the network and another set S from which vertices may be added [9, 13, 20, 21, 23, 42, 47, 62, 86]. This fact is called Thue’s Theorem but had been shown for lattices already by Lagrange in 1773 and a complete proof is due to Fejes Tóth. Community content is available under CC BY-NC-SA unless otherwise noted. Clearly, for any packing to be possible, the sum of. [GW1]) had by itsThe Tóth Sausage Conjecture: 200 creat 200 creat Tubes within tubes within tubes. AbstractLet for positive integersj,k,d and convex bodiesK of Euclideand-spaceEd of dimension at leastj Vj, k (K) denote the maximum of the intrinsic volumesVj(C) of those convex bodies whosej-skeleton skelj(C) can be covered withk translates ofK. Introduction. Radii and the Sausage Conjecture. Toth’s sausage conjecture is a partially solved major open problem [2]. Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inE d ,n be large. 3. . Wills (2. For the pizza lovers among us, I have less fortunate news. 1. Instead, the sausage catastrophe is a mathematical phenomenon that occurs when studying the theory of finite sphere packing. It remains an interesting challenge to prove or disprove the sausage conjecture of L. Ball-Polyhedra. §1. dot. Feodor-Lynen Forschungsstipendium der Alexander von Humboldt-Stiftung. If you choose the universe next door, you restart the. The sausage conjecture holds in \({\mathbb{E}}^{d}\) for all d ≥ 42. • Bin packing: Locate a finite set of congruent balls in the smallest volume container of a specific kind. J. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. The cardinality of S is not known beforehand which makes the problem very difficult, and the focus of this chapter is on a better. Currently, the sausage conjecture has been confirmed for all dimensions ≥ 42. Further he conjectured Sausage Conjecture. 4. 3 (Sausage Conjecture (L. Trust governs how many processors and memory you have, which in turn govern the rate of operation/creativity generation per second and how many maximum operations are available at a given time (respectively). Accepting will allow for the reboot of the game, either through The Universe Next Door or The Universe WithinIn higher dimensions, L. Message from the Emperor of Drift is unlocked when you explore the entire universe and use all the matter to make paperclips. Conjecture 1. Conjecture 2. Let k non-overlapping translates of the unit d -ball B d ⊂E d be given, let C k be the convex hull of their centers, let S k be a segment of length 2 ( k −1) and let V denote the volume. In this. Fejes Toth conjectured that in E d , d ≥ 5, the sausage arrangement is denser than any other packing of n unit balls. Containment problems. Show abstract. An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given. The Sausage Catastrophe (J. , Bk be k non-overlapping translates of the unid int d-bal euclideal Bn d-space Ed. Max. For ϱ>0 the density δ (C;K,ϱ) is defined by δ(C;K,ϱ) = card C·V(K)/V(conv C+ϱK). Fejes Tóth) states that in dimensions d ≥ 5, the densest packing of any finite number of spheres in R^d occurs if and only if the spheres are all packed in a line, i. Toth’s sausage conjecture is a partially solved major open problem [2]. In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. Furthermore, led denott V e the d-volume. Conjecture 1. Trust is gained through projects or paperclip milestones. László Fejes Tóth, a 20th-century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit spheres. ) but of minimal size (volume) is looked The Sausage Conjecture (L. Toth’s sausage conjecture is a partially solved major open problem [2]. Fejes Toth, Gritzmann and Wills 1989) (2. The sausage conjecture for finite sphere packings of the unit ball holds in the following cases: 870 dimQ<^(d-l) P. AbstractIn 1975, L. 3 (Sausage Conjecture (L. Abstract In this note we present inequalities relating the successive minima of an $o$ -symmetric convex body and the successive inner and outer radii of the body. BETKE, P. Fejes Tóth's sausage conjecture then states that from = upwards it is always optimal to arrange the spheres along a straight line. ( 1994 ) which was later improved to d ≥. Quên mật khẩuup the idea of Zong’s proof in [11] and show that the “spherical conjecture” is also valid in Minkowski Geometry. Fejes T oth [25] claims that for any number of balls, a sausage con guration is always best possible, provided d 5. In particular we show that the facets ofP induced by densest sublattices ofL 3 are not too close to the next parallel layers of centres of balls. Math. It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. Acceptance of the Drifters' proposal leads to two choices. There are 6 Trust projects to be unlocked: Limerick, Lexical Processing, Combinatory Harmonics, The Hadwiger Problem, The Tóth Sausage Conjecture and Donkey Space. Fejes Toth's sausage conjecture 29 194 J. The sausage conjecture holds for convex hulls of moderately bent sausages B. BOKOWSKI, H. N M. Usually we permit boundary contact between the sets. Ulrich Betke works at Fachbereich Mathematik, Universität Siegen, D-5706 and is well known for Intrinsic Volumes, Convex Bodies and Linear Programming. Close this message to accept cookies or find out how to manage your cookie settings. 1016/0012-365X(86)90188-3 Corpus ID: 44874167; An application of valuation theory to two problems in discrete geometry @article{Betke1986AnAO, title={An application of valuation theory to two problems in discrete geometry}, author={Ulrich Betke and Peter Gritzmann}, journal={Discret. 1 [[quoteright:350:2 [[caption-width-right:350:It's pretty much Cookie Clicker, but with paperclips. jeiohf - Free download as Powerpoint Presentation (. Đăng nhập . FEJES TOTH'S SAUSAGE CONJECTURE U. Click on the article title to read more. [9]) that the densest pack ing of n > 2 unit balls in Ed, d > 5, is the sausage arrangement; namely the centers are collinear. M. Fejes. org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. , among those which are lower-dimensional (Betke and Gritzmann 1984; Betke et al. GRITZMANN AND J. for 1 ^ j < d and k ^ 2, C e . Sausage Conjecture 200 creat 200 creat Tubes within tubes within tubes. Tóth’s sausage conjecture is a partially solved major open problem [3]. Further lattic in hige packingh dimensions 17s 1 C. BeitrAlgebraGeom as possible”: The first one leads to so called bin packings where a container (bin) of a prescribed shape (ball, simplex, cube, etc. Furthermore, led denott V e the d-volume. This has been known if the convex hull Cn of the centers has low dimension. 15. 453 (1994) 165-191 and the MathWorld Sausage Conjecture Page). Toth’s sausage conjecture is a partially solved major open problem [2]. Acceptance of the Drifters' proposal leads to two choices. Fejes Tóths Wurstvermutung in kleinen Dimensionen" by U. Ulrich Betke | Discrete and Computational Geometry | We show that the sausage conjecture of Laszlo Fejes Toth on finite sphere packings is true in dimens. Fejes Tóth, 1975)). Math. L. It is also possible to obtain negative ops by using an autoclicker on the New Tournament button of Strategic Modeling. It appears that at this point some more complicated. Gritzmann, J. Fejes Tóth's sausage conjecture - Volume 29 Issue 2. An upper bound for the “sausage catastrophe” of dense sphere packings in 4-space is given. Discrete & Computational Geometry - We show that the sausage conjecture of László Fejes Tóth on finite sphere packings is true in dimension 42 and above. In this. The sausage conjecture appears to deal with a simple problem, yet a proof has proved elusive. Download to read the full article text Working on a manuscript? Avoid the common mistakes Author information. In -D for the arrangement of Hyperspheres whose Convex Hull has minimal Content is always a ``sausage'' (a set of Hyperspheres arranged with centers along a line), independent of the number of -spheres. ) but of minimal size (volume) is lookedMonatsh Math (2019) 188:611–620 Minimizing the mean projections of finite ρ-separable packings Károly Bezdek1,2. and the Sausage Conjectureof L. A zone of width ω on the unit sphere is the set of points within spherical distance ω/2 of a given great circle. WILLS ABSTRACT Let k non-overlapping translates of the unit d-ball B d C E a be given, let Ck be the convex hull of their centers, let Sk be a segment of length 2(k - 1) and let V denote the volume. Packings and coverings have been considered in various spaces and on. Polyanskii was supported in part by ISF Grant No. Wills. That is, the sausage catastrophe no longer occurs once we go above 4 dimensions. L. 266 BeitrAlgebraGeom(2021)62:265–280 as possible”: The first one leads to so called bin packings where a container (bin) of a prescribed shape (ball, simplex, cube, etc. Abstract. Letk non-overlapping translates of the unitd-ballBd⊂Ed be given, letCk be the convex hull of their centers, letSk be a segment of length 2(k−1) and letV denote the volume. Costs 300,000 ops. The conjecture was proposed by Fejes Tóth, and solved for dimensions >=42 by Betke et al. BRAUNER, C. In higher dimensions, L. BRAUNER, C. 4. Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inEd,n be large. In 1998 they proved that from a dimension of 42 on the sausage conjecture actually applies. Fejes Toth made the sausage conjecture in´It is proved that for a densest packing of more than three d -balls, d geq 3 , where the density is measured by parametric density, the convex hull of their centers is either linear (a sausage) or at least three-dimensional. However, instead of occurring at n = 56, the transition from sausages to clusters is conjectured to happen only at around 377,000 spheres. Radii and the Sausage Conjecture. Mathematika, 29 (1982), 194. Enter the email address you signed up with and we'll email you a reset link. In such27^5 + 84^5 + 110^5 + 133^5 = 144^5. If the number of equal spherical balls. W. A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball B d of the Euclidean d-dimensional space E d can be packed ([5]). Manuscripts should preferably contain the background of the problem and all references known to the author. Conjecture 2. V. The dodecahedral conjecture in geometry is intimately related to sphere packing. Pachner, with 15 highly influential citations and 4 scientific research papers. Expand. 1953. . For the sake of brevity, we will say that the pair of convex bodies K, E is a sausage if either K = L + E where L ∈ K n with dim L ≤ 1 or E = L + K where L ∈ K n with dim L ≤ 1. Fejes Toth conjectured that in Ed, d ≥ 5, the sausage arrangement is denser than any other packing of n unit balls. Finite and infinite packings. Đăng nhập bằng google. However the opponent is also inferring the player's nature, so the two maneuver around each other in the figurative space, trying to narrow down the other's. 1 (Sausage conjecture:). org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. 1 (Sausage conjecture) Fo r d ≥ 5 and n ∈ N δ 1 ( B d , n ) = δ n ( B d , S m ( B d )). Slices of L. 4 A. This has been known if the convex hull C n of the centers has. • Bin packing: Locate a finite set of congruent spheres in the smallest volume containerIn this work, we confirm this conjecture asymptotically by showing that for every (varepsilon in (0,1]) and large enough (nin mathbb N ) a valid choice for this constant is (c=2-varepsilon ). However Wills ([9]) conjectured that in such dimensions for small k the sausage is again optimal and raised the problemIn this survey we give an overview about some of the main results on parametric densities, a concept which unifies the theory of finite (free) packings and the classical theory of infinite packings. 2 Pizza packing. The meaning of TOGUE is lake trout. . SLOANE. Fejes Tóth and J. The Tóth Sausage Conjecture is a project in Universal Paperclips. An approximate example in real life is the packing of. Casazza; W. BAKER. It follows that the density is of order at most d ln d, and even at most d ln ln d if the number of balls is polynomial in d. " In. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. That is, the shapes of convex bodies containing m translates of a convex body K so that their Minkowskian surface area is minimum tends to a convex body L. math. ” Merriam-Webster. Kuperburg, On packing the plane with congruent copies of a convex body, in [BF], 317–329; MR 88j:52038. Fejes T´oth ’s observation poses two problems: The first one is to prove or disprove the sausage conjecture. It is available for the rest of the game once Swarm Computing is researched, and it supersedes Trust which is available only during Stage 1. For d = 2 this problem was solved by Groemer ([6]). homepage of Peter Gritzmann at the. Contrary to what you might expect, this article is not actually about sausages. F. The Universe Next Door is a project in Universal Paperclips. In suchRadii and the Sausage Conjecture. svg","path":"svg/paperclips-diagram-combined-all. That is, the shapes of convex bodies containing m translates of a convex body K so that their Minkowskian surface area is minimum tends to a convex body L. L. The length of the manuscripts should not exceed two double-spaced type-written. Slices of L. In n dimensions for n>=5 the arrangement of hyperspheres whose convex hull has minimal content is always a "sausage" (a set of hyperspheres arranged with centers along a line), independent of the number of n-spheres. László Fejes Tóth, a 20th-century Hungarian geometer, considered the Voronoi decomposition of any given packing of unit spheres. 6. L. 16:30–17:20 Chuanming Zong The Sausage Conjecture 17:30 in memoriam Peter M. improves on the sausage arrangement. View details (2 authors) Discrete and Computational Geometry. J. Thus L. Based on the fact that the mean width is. The overall conjecture remains open. See also. 19. The sausage catastrophe still occurs in four-dimensional space. 5 The CriticalRadius for Packings and Coverings 300 10. BeitrAlgebraGeom as possible”: The first one leads to so called bin packings where a container (bin) of a prescribed shape (ball, simplex, cube, etc. Anderson. In 1975, L. Full text. Further he conjectured Sausage Conjecture. Semantic Scholar's Logo. The Tóth Sausage Conjecture is a project in Universal Paperclips. Slice of L Feje. L. DOI: 10. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. 275 +845 +1105 +1335 = 1445. SLICES OF L. View. A. The Universe Within is a project in Universal Paperclips. Fejes Tóth's sausage conjecture, says that ford≧5V. Sausage Conjecture. (1994) and Betke and Henk (1998). Doug Zare nicely summarizes the shapes that can arise on intersecting a. Click on the title to browse this issueThe sausage conjecture holds for convex hulls of moderately bent sausages @article{Dekster1996TheSC, title={The sausage conjecture holds for convex hulls of moderately bent sausages}, author={Boris V. But it is unknown up to what “breakpoint” be-yond 50,000 a sausage is best, and what clustering is optimal for the larger numbers of spheres. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. Keller conjectured (1930) that in every tiling of IRd by cubes there are twoSemantic Scholar profile for U. F. Fejes Toth conjectured 1. Consider an arrangement of $n$ congruent zones on the $d$-dimensional unit sphere $S^{d-1}$, where a zone is the intersection of an origin symmetric Euclidean plank. Abstract. Wills, SiegenThis article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. PACHNER AND J. J. N M. The Simplex: Minimal Higher Dimensional Structures. 3 (Sausage Conjecture (L. Fejes Toth, Gritzmann and Wills 1989) (2. A basic problem in the theory of finite packing is to determine, for a given positive integer k , the minimal volume of all convex bodies into which k translates of the unit ball B d of the Euclidean d -dimensional space E d can be packed ([5]). H. We present a new continuation method for computing implicitly defined manifolds. Expand. V. Here we optimize the methods developed in [BHW94], [BHW95] for the special A conjecture is a statement that mathematicians think could be true, but which no one has yet proved or disproved. The second theorem is L. It is a problem waiting to be solved, where we have reason to think we know what answer to expect. 4 A. CONWAYandN. , a sausage. 99, 279-296 (1985) Mathemalik 9 by Springer-Verlag 1985 On Two Finite Covering Problems of Bambah, Rogers, Woods and ZassenhausIntroduction. The. A basic problem in the theory of finite packing is to determine, for a given positive integer k, the minimal volume of all convex bodies into which k translates of the unit ball B d of the Euclidean d-dimensional space E d can be packed ([5]). . Further o solutionf the Falkner-Ska. All Activity; Home ; Philosophy ; General Philosophy ; Are there Universal Laws? Can you break them?Diagrams mapping the flow of the game Universal Paperclips - paperclips-diagrams/paperclips-diagram-stage2. Introduction. The action cannot be undone. A new continuation method for computing implicitly defined manifolds is presented, represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a bounda. From the 42-dimensional space onwards, the sausage is always the closest arrangement, and the sausage disaster does not occur. Fejes Tóth for the dimensions between 5 and 41. The Spherical Conjecture The Sausage Conjecture The Sausage Catastrophe Sign up or login using form at top of the. inequality (see Theorem2). 1. LAIN E and B NICOLAENKO. Gritzmann, P. a sausage arrangement in Ed and observed δ(Sd n) <δ(d) for all n, provided that the dimension dis at least 5. In 1975, L. FEJES TOTH'S "SAUSAGE-CONJECTURE" BY P. Khinchin's conjecture and Marstrand's theorem 21 248 R. This definition gives a new approach to covering which is similar to the approach for packing in [BHW1], [BHW2]. It is proved that for a densest packing of more than three d -balls, d geq 3 , where the density is measured by parametric density, the convex hull of their centers is either linear (a sausage) or at least three-dimensional. This has been. WILLS. 1This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. • Bin packing: Locate a finite set of congruent balls in the smallest volume container of a specific kind. space and formulated the following conjecture: for n ~ 5 the volume of the convex hull of k non-overlapping unit balls attains its minimum if the centres of the balls are equally spaced on a line with distance 2, so that the convex hull of the balls becomes a "sausage". Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inEd,n be large. J.