Abstract
Theorem. Let a set X⊂Rn have unit circumradius and let B be the unit ball containing X. Put C =conv \(\bar X\) D =diam C (=diam X), k =dim C,d i = √(2i + 2)/i. Then: (i) D∈[dn, 2]; (ii) k≧m where m∈{2,3,...,n} satisfies D∈[dm, dm−1) (di decreases by i); (iii) In case k=m (by (ii), this is always the case when m=n), C contains a k-simplex Δ such that: (α) its vertices are on δB; (β) the centre of B belongs toint Δ; (γ) the inequalitiesλ k (D) ≦l ≦D with
are unimprovable estimates for length l of any edge of Δ.
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Supported by a Canadian NSERC Grant.
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Dekster, B.V. An extension of Jung’s theorem. Israel J. Math. 50, 169–180 (1985). https://doi.org/10.1007/BF02761397
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DOI: https://doi.org/10.1007/BF02761397