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 Dec23 comment Does this operation result in a convex set Sorry. I cannot see why $(1-t)a_1 + t a_2$ is a point of $X$. Dec23 asked Does this operation result in a convex set Aug24 awarded Scholar Aug24 accepted interchange summation and iterated integration Aug24 comment interchange summation and iterated integration Thank all, Evan, Cameron, and @T.Bongers. Aug24 asked interchange summation and iterated integration Apr21 revised circular table game added 49 characters in body Apr19 comment circular table game I don't know. In quote is the exact problem description. @Abel Apr19 revised circular table game added 151 characters in body Apr19 asked circular table game Feb14 comment the segment through a fixed interior point of a compact convex set which is at least as long as its parallels Thanks. I guess that a proof without relating to homotopy and vector field is expected. The problem is an exercise just after the concept of convexity and affinity, the Carathéodory theorem and the separation theorem are presented in the book. Feb10 comment the segment through a fixed interior point of a compact convex set which is at least as long as its parallels I understand the "it suffices" part. But it is still puzzling to me how to deduce the conclusion using the theorem aforementioned. Could anyone give more details? @5PM Feb10 awarded Supporter Feb10 awarded Teacher Feb10 awarded Editor Feb10 revised What does it mean when you say that the function is bounded? added 84 characters in body Feb10 awarded Student Feb10 comment What does it mean when you say that the function is bounded? Not really. See what if $f(n)=n$ and $g(n) = n^2$ where $n\in\mathbb{N}^*$. Feb10 answered What does it mean when you say that the function is bounded? Feb10 asked the segment through a fixed interior point of a compact convex set which is at least as long as its parallels