An extremal property of random flats

This paper deals with anisotropic random two‐dimensional flats meeting a ball B in the four‐dimensional Euclidean space. The probability p that two independent, identically distributed random flats have a common point in B depends on the direction distribution of the random flats. The direction distributions maximizing p are determined. If the random flats are isotropic, p is not maximal, in contrast to the case of random hyperplanes (three‐dimensional flats). Related results for flat processes are explained.