A beautiful construction by Besicovitch and Rado [1] produces an astounding example of a compact connected plane set of measure zero containing circles of all radii $r\in(0,1]$.
A corollary to a theorem proved in [2] states that there exists a nowhere-dense compact connected plane set containing a translated image of the boundary of every compact convex set of diameter at most one.
Question: Is there a compact connected plane set of measure zero containing a translated image of the boundary of every compact convex set of diameter at most one?
[1] Besicovitch, A. S.; Rado, R., J. London Math. Soc. 43 (1968) 717–719, MR0229779
[2] Holsztyński, W.; Kuperberg, W.; Mycielski, J., Houston J. Math. 3 (1977), no. 4, 475–476, MR0461433