Abstract
It is a well known result of Borsuk [21 that any mapping2 of the n-sphere Sn into euclidean n-space carries some two antipodal points of S,, into the same point; that is, the inverse of some point has the same diameter as Sn. It is the principal purpose of this paper to obtain results concerning the diameter of connected subsets of such inverse sets when convex sets are mapped into spaces of lower dimension. The first part of the paper concerns itself with convex sets of dimension greater than two; in the second part somewhat sharper results are obtained for mappings of plane convex sets.
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