Abstract
In a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r − 1)+1 points inR dhas anr-partition into (pair wise disjoint) subsetsS =S 1 ∪ … ∪S r so that\(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS i # Ø. This note considers the following more general problems: (1) How large mustS σR dbe to assure thatS has anr-partitionS=S 1∪ … ∪S r so that eachn members of the family {convS i ∼ ri-1 have non-empty intersection, where 1<=n<=r. (2) How large mustS ∪R d be to assure thatS has anr-partition for which\(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS r is at least 1-dimensional.
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Reay, J.R. Several generalizations of Tverberg’s theorem. Israel J. Math. 34, 238–244 (1979). https://doi.org/10.1007/BF02760885
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DOI: https://doi.org/10.1007/BF02760885