On Two Conjectures of Steinhaus

We disprove two conjectures of H. Steinhaus by showing that: (1) there is a convex surface S such that for any point x on S and any point y in the set Fx of farthest points from x, there are at most two segments from x to y; (2) the properties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left| {Fx} \right| = 1$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${F_{F_x } = x}$$ \end{document}do not characterize the sphere.