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Oct 27, 2018 at 19:44 comment added MasM You argued about non-intersecting situations and it seems majority of them are related to convex bodies in their non-smooth points, it is still interesting to figure out the competition between amount and treat of intersecting and non-intersecting points in more complex shapes than just convex and over different angles of rotation, for example finding shapes that your set $R(\alpha)$ becomes dense and uncountable.
Aug 5, 2018 at 22:21 comment added MasM See this question: mathoverflow.net/questions/306703/… it is somehow related to this problem and your answer.
Jul 21, 2018 at 23:03 vote accept MasM
Jul 21, 2018 at 23:06
Jul 9, 2018 at 13:51 comment added erz @MasM This does seem relevant indeed, but I don't see how yet.
Jul 9, 2018 at 13:20 comment added MasM It seems that I have found something important about this problem, check the third answer
May 13, 2018 at 20:47 history edited erz CC BY-SA 4.0
added 1083 characters in body
May 13, 2018 at 1:19 history answered erz CC BY-SA 4.0