On generalizations of Radon's theorem and the Ham sandwich theorem (Q1260776)

The authors raise the following interesting conjecture. Let \(0\leq k\leq d- 1\) and let \(S_ 0,\dots,S_ k\) be finite sets of points in \(\mathbb{R}^ d\), with \(| S_ i|=(r_ i-1)\) \((d-k+1)+1\) for \(i=0,\dots,k\). Then \(S_ i\) can be split into \(r_ i\) sets, \(S_{i1}\), \(S_{i2},\dots,S_{ir_ i}\), so that there is a \(k\)-flat \(F\) meeting all the sets conv \(S_{ij}\), \(0\leq i\leq k\), \(1\leq j\leq r_ i\). This conjecture would imply a theorem of \textit{R. T. Zivaljević} and the second author [Bull. Lond. Math. Soc. 22, No. 2, 183-186 (1990; Zbl 0709.60011)]. Some of its special cases are proved here. Further, the paper contains a new and simple proof of Tverberg's theorem [\textit{H. Tverberg}, J. Lond. Math. Soc. 41, 123-128 (1966; Zbl 0131.20002)].