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This means that in dimensions 2 and 3, when m is large, the sausage arrangement does not minimize the volume of a convex hull which contains m unit balls. However U. Betke, M. M. Wills [1] were recently able confirm the sausage conjecture for very high dimensions. Denote the family of all those convex bodies of minimal volume containing m disjoint unit balls by Bo,m. The L. Fejes T6th-Betke-Henk-Wills Phenomenon: Among all the n-dimensional convex bodies Bm which contain m nonoverlapping unit balls, 24 2.

For z E Z, let z denote the element z + 9Z of Zg. Taking 10 Ai = U{(zl,Z2, ... , ,ZlO) E Z§o: zi = -4, ... ,4; zj = 0, j -=I- i}, i=l it is easy to see that card{Ai} = 81. Let A:;; = U. Since U is a perfect cover of Z§o, Z§o is a product of Ai and A:;;. Let f be the homomorphism f : ZlO 1---+ Z§o defined by f(Zl,Z2, ... ,ZlO) = (Zl,z2, ... ,ZlO) and take Ai = 1(4,10) C ZlO. 7, ZlO is a product of Ai and f-l(A:;;). Thus, M(4, 10), and hence also M(4, 10, t, 8), tiles RlO with X = f-l(A:;;). Next, we show that M(4, 10, t, 8) is not a lattice tile.

Suppose, to the contrary, that there exists a face F with dim(F) = rand two distinct points Xl, X2 E X(o, F) such that int(P+xl)nint(P+x2) #- 0. Then by the definition of X(o, F), we can find 1 points Yl = Xl, Y2, ... , YI = X of X(o, F) such that each (P + Yi) n (P + Yi+d, containing F, is a common facet of P+Yi and P+Yi+l. In this situation, we call the sequence P + Yl, P + Y2, ... , P + YI an {Xl, x2}-chain. 4, then this {Xl, x2}-chain gives rise to a corresponding {Xl, X2} * -chain of spherical (n - r - 1) -polytopes Pt = (P + Yi) n B* for i = 1,2, ...

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