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Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of area bisectors of planar convex regions in 3d.

We can consider several types of bisections of 3D convex solids - cutting the solid by a plane into 2 convex pieces of equal volume, equal surface area, equal width and so on..The intersection of a bisecting plane with the body can be called a bisecting section. Naturally, a solid has volume bisecting sections, surface bisecting sections and so forth.

So, an all-in one algorithmic question is:

  • Given a convex polyhedron, how does one find the volume bisecting section (equally surface area bisecting section, etc) with largest (equally, smallest) area (equally, perimeter)?

Further, for general convex solids:

  • if all volume (surface area) bisecting sections of a solid have same area (perimeter), what could we infer?

  • if all volume (surface area) bisecting sections of a solid pass through a point, is the solid centrally symmetric?

Note: more speculatively, we can cask: do we have some relations between bisecting sections like (say) "for any convex solid, the volume bisecting section with least area is the same as the volume bisecting section with least perimeter" or "the volume bisecting section with max area is the same as the surface area bisecting section with max perimeter"?

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    $\begingroup$ For a polyhedron you could easily imagine a discrete algorithm. Take an orientation of the polyhedron and move it vertically such that the plane $z=0$ bisects your polyhedron. You should be able to find algorithms computing all geometric quantities for the bisecting sections. These algorithms should be quick, since only geometric quantities are involved. Since differentiating these objective functions might be tricky, you could use gradient free optimization (Golden Search in 1D, Nelder in 3D) for the optimization part. $\endgroup$
    – Beni Bogosel
    Commented Jun 15 at 10:11
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    $\begingroup$ Concerning the second question, do you have a conjecture? The 2D analogue is solved? $\endgroup$
    – Beni Bogosel
    Commented Jun 15 at 10:12
  • $\begingroup$ The only possibility one could think of is the solid being a sphere. For example, “if all volume bisecting sections have same area, the solid is a sphere”. Basically one was trying to see the various 3d analogs of the Reuleaux triangle. A 2d analog qn is linked above now. Thanks. $\endgroup$
    – Nandakumar R
    Commented Jun 15 at 11:08
  • $\begingroup$ If the 2D question is not yet solved, I doubt there's a quicker idea in 3D... $\endgroup$
    – Beni Bogosel
    Commented Jun 15 at 11:20
  • $\begingroup$ The 2d version seems done and closed now after an answer came in on zinder curves. 3d analogs of zinder curves seem explored but with width rather than properties of sections as the defining property. $\endgroup$
    – Nandakumar R
    Commented Jun 30 at 6:04

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