It is shown that among all cumulative distribution functions passing through $k \geqq 2$ given points there is a unique one with minimal Fisher information; it is obtained by a curious type of spline interpolation. This answers some questions raisd by D. G. Kendall and J. W. Tukey.
"Fisher Information and Spline Interpolation." Ann. Statist. 2 (5) 1029 - 1033, September, 1974. https://doi.org/10.1214/aos/1176342822