Ref: 1. A claim on planar sections of 3D convex bodies
Given a 3D convex body $C$ and a specified direction $n$, we consider the planar sections of $C$ normal to $n$ with (1) maximum diameter and (2) maximum least width.
- How does one characterize those convex bodies for which for every $n$, the diameter of the corresponding max diameter section is the same?
(Following a comment in ref 2 above from Yoav Kallus, an oblate spheroid is an example of such a body as are bodies of constant width).
Same question as above with maximum least width replacing of max diameter. (a prolate spheroid is an example of such a body).
What about bodies for which for every $n$, the ratio between the diameter of the max diameter section and least width of the max least width section (or the maximum from among sections normal to $n$ of the ratio between diameter and least width) is a constant?
