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This is an extension of On segments of equal area cut from planar convex regions by chords.

While the 3D analog of the above question would be about 3D pieces cut from a convex body $C$ by planes, here we ask about planar sections.

Question: Other than the solid sphere, is there any 3D convex body of unit volume $C$ with the property: all equal area planar sections of $C$ with even one specified area $\alpha$ also have the same perimeter (we refer here to the area and perimeter of the section itself)?

As pointed out by Jukka Kohonen below, when the area of cross section $\alpha$ is maximal, examples of such solids do exist; so the question probably ought to really focus on non-maximal values of $\alpha$.

And does this question have any connection to surfaces of constant width or bodies of const brightness? Another earlier post in the same ballpark: A claim on planar sections of 3D convex bodies.

Note: we could replace "equal perimeter" with "equal diameter" or say, "are at equal perpendicular distance from some point" etc... (in general, " all sections with quantity A also having quantity B equal") to get other questions.

Nandakumar R
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Yes, with the unit cube and $\alpha = \sqrt{2}$, which is the maximum possible area of a cross-section. This area is attained only with rectangles with two sides of length $1$ (coinciding with the cube's edges) and two sides of length $\sqrt{2}$. So perforce those cross-sections have the same perimeter, namely $2+2\sqrt{2}$.

There are six such rectangles, so the solution is not trivial in the sense of having only one unique maximal-area cross section. I guess there would be other shapes with a similar situation, if you set $\alpha$ to be the maximal cross-section area. If $\alpha$ is not maximal, the question becomes more challenging.

That the area $\sqrt{2}$ is maximal in the unit cube, and only attained by these rectangles, follows from Ball's theorem:

Every section of the unit cube, $Q_n$, by an $(n-1)$-dimensional subspace, $H$, has volume at most $\sqrt{2}$. This upper bound is attained if and only if $H$ contains an $(n-2)$-dimensional face of $Q_n$.

Ball, Keith, Cube slicing in $\mathbf{R}^ n$, Proc. Am. Math. Soc. 97, 465-473 (1986). ZBL0601.52005.

Jukka Kohonen
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  • Thanks. What I had in mind were indeed values of alpha such that a convex body has infinitely many cross sections with same area alpha. I guess what you showed by example is a sort of extreme - say degenerate - situation. Indeed, even in https://mathoverflow.net/questions/457094/on-segments-of-equal-area-cut-from-planar-convex-regions-by-chords, the comment by Pierre about centrally symmetric planar regions indicates a similar situation I guess. – Nandakumar R Oct 28 '23 at 17:54
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    For a continuum of sections, consider the right circular cylinder of diameter $10$ and height $1$. The largest cross-sections would be the ellipses that touch each base at a point; since they are congruent, they have the same perimeter. Perhaps you want to restrict $\alpha$ to be nonmaximal, or perhaps you want incongruent cross sections? – Jukka Kohonen Oct 28 '23 at 19:01
  • Thanks again. It probably needs to be mentioned that alpha is non-maximal; shall edit question. There seems to be any need to insist that the cross sections should be incongruent but the non-maximality of alpha is I think important. – Nandakumar R Oct 28 '23 at 19:33