Analogues of Horn’s theorem for finite unions of starshaped sets in ℝd

Fix k, d, 1 ≤ k ≤ d + 1. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F} $$\end{document} be a nonempty, finite family of closed sets in ℝd, and let L be a (d − k + 1)-dimensional flat in ℝd. The following results hold for the set T ≡ ∪{F: F in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F} $$\end{document}}.