Here is an illustration of Gerry Myerson's nice idea: <br /> ![onions][1] <br /> The left set has onion depth $n/3$, the right set, after small rotations, has depth 1. Incidentally, there is an efficient algorithm to find the onion depth of a point set: $O( n \log n )$ for a set of $n$ points, established by Bernard Chazelle in the paper, "On the convex layers of a planar set," [IEEE Transactions on Information Theory, 31: 509-517, 1985][2]; doi: [10.1109/TIT.1985.1057060](https://doi.org/10.1109/TIT.1985.1057060), [semanticscholar](https://pdfs.semanticscholar.org/ba44/999337850b5dd5a03171da57c2712bcbc1e5.pdf). There also has been some work on the combinatorial structure of onion layers. A crude summary is: the structure is complex and not well understood. See "Onion polygonizations," [_Information Processing Letters_ Volume 57, Issue 3, 12 February 1996, Pages 165-173][3], doi: [10.1016/0020-0190(95)00193-X](https://doi.org/10.1016/0020-0190(95)00193-X). [1]: https://i.sstatic.net/mwqEA.jpg [2]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.8709 [3]: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0F-3VSNJ4W-X&_user=825413&_coverDate=02%252F12%252F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1593169683&_rerunOrigin=google&_acct=C000044660&_version=1&_urlVersion=0&_userid=825413&md5=bca2cfa79aeef0fece644ecb8d749466&searchtype=a