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We add a bit more on shadows and planar sections following On a pair of solids with both corresponding maximal planar sections and shadows having equal area . We consider only polyhedrons.

  1. Given a convex polyhedral solid, how does one find its largest planar section? 'largest' could mean max area or max perimeter. So, this is 2 questions in 1.

If one could prove that either or both largest planar sections necessarily pass through 2 vertices (say), that could be useful.

  1. To find an algorithm that finds the largest shadow of a given convex polyhedral solid (again area and perimeter versions).

Again, if one could prove some 'lemma' regarding vertices of the polyhedron, it could be nice.

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Your Question 2 (max/min area/volume shadow version) is answered in this paper:

McKenna, Michael, and Raimund Seidel. "Finding the optimal shadows of a convex polytope." In Proceedings 1st Annual Symposium on Computational Geometry (SoCG), pp. 24-28. 1985.

The algorithms (there are two) have time-complexity $O^*(n^{d-1})$ for $n$ vertex polytopes in dimension $d$, with $\mbox{}^*$ meaning with our without a $\log n$ factor. So quadratic time in $d=3$.

Here's a screenshot from the Introduction:

Intro

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