Sausages are good packings |
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Authors: | U Betke M Henk J M Wills |
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Institution: | 1. Math. Institut, Universit?t Siegen, D-57068, Siegen, Germany 2. Math. Institut, Technische Universit?t Berlin, D-10623, Berlin, Germany
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Abstract: | LetB
d
be thed-dimensional unit ball and, for an integern, letC
n
={x
1,...,x
n
} be a packing set forB
d
, i.e.,|x
i
−x
j
|≥2, 1≤i<j≤n. We show that for every
a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C
n
)+ρB
d
)≥V(conv(S
n
)+ρB
d
), whereS
n
is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove
that, for every convex bodyK and ρ<1/32d
−2,V(conv(C
n
)+ρK)≥V(conv(S
n
)+ρK), whereC
n
is a packing set with respect toK andS
n
is a minimal “sausage” arrangement ofK, holds. |
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Keywords: | |
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