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Partly convex Peano curves. (English) Zbl 0545.26003

The following theorem is proved: There exists a continuous function from the unit interval \([0,1]\) onto the unit square such that the image of each initial segment \([0,\alpha]\) and of each final segment \([\alpha,1]\) is a convex set. The original question of Mihalik and Wieczorek whether the same assertion is true if the image of each subinterval \([\alpha,\beta]\subseteq [0,1]\) needs to be convex remains unsolved. However, a discrete version of this conjecture is verified: For every natural number n, there exists a sequence \(C_ 1^{(n)},C_ 2^{(n)},...,C_ N^{(n)}\) of compact convex sets satisfying \((i)\quad diam C_ r^{(n)}\leq 1\) for every \(1\leq r\leq N, (ii)\quad \cup^{s}_{i=r}C_ i^{(n)}\) is convex for every \(1\leq r<s\leq N, (iii)\quad \cup^{N}_{i=1}C_ i^{(n)}\) contains a disc of radius n.

MSC:

26A30 Singular functions, Cantor functions, functions with other special properties
52A10 Convex sets in \(2\) dimensions (including convex curves)
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