I also Think The Kakeya Needle Problem is worth mentioning (see http://mathworld.wolfram.com/KakeyaNeedleProblem.htmlhttps://mathworld.wolfram.com/KakeyaNeedleProblem.html). To me it is counter-intuitive that there is no smallest set, in which a needle of unit length can be freely rotated. Unless it has to be convex, of course.