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This post follows Are two convex solids with all corresponding shadows equal in area congruent?

Every convex 3D body has planar sections with normals in any given direction. We consider the maximum area planar section (there may be many such sections) of a convex body in each direction.

  1. If 2 convex bodies P and Q can be placed such that the max area planar section of P in any given direction has equal area to the max area planar section of Q in that direction, can we say that P and Q have same volume/surface area? What else could we infer? Will restricting to convex polyhedrons have any implication?

  2. If P and Q are such that the max perimeter planar section of P in any direction has same perimeter as the max perimeter section of Q in that direction, what could be said?

  3. And what are the implications of both above properties - equality of areas and perimeters - holding for a pair of solids P and Q?

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    $\begingroup$ You can find an answer to some of your questions in even dimensions in arxiv.org/abs/1112.3976 . Unfortunately, $3$ is odd, which creates a lot of trouble with that approach. $\endgroup$
    – fedja
    Commented Jun 6 at 22:56
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    $\begingroup$ And arxiv.org/abs/1201.0393 is specifically about maximal sections (now in all dimensions including 3) $\endgroup$
    – fedja
    Commented Jun 6 at 23:11
  • $\begingroup$ thank you. i understand that theorem 2 in the first reference you have answers the area case of the problem in even dimensions - even if every max section and projection of body 1 equal in area a corresponding max section - projection pair of body 2, the two bodies need not be congruent. this seems to answer the first question in mathoverflow.net/questions/472548/…. maybe the 'perimeter case' of these questions hasn't been explored. $\endgroup$
    – Nandakumar R
    Commented Jun 8 at 10:35

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