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We add a bit to More on shadows of 3D convex bodies

By a shadow of a 3D body, we mean the orthogonal projection of it onto a 2D plane.

  • If all shadows of a convex 3D body have the same diameter, will the body necessarily be one of constant width?

The least width of a planar figure is the perpendicular distance between the closest pair of parallel lines that are tangent to it.

  • If all shadows of a convex 3D body have the same least width, will the body necessarily be one of constant width? And what if for all shadows, the ratio between the diameter and least width is the same?

Note: Further variants of this question could be asked with other quantities instead of diameter and least width.

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    $\begingroup$ think about the case of oblate and prolate spheroids for the two questions $\endgroup$
    – Yoav Kallus
    Commented Aug 8, 2023 at 14:10
  • $\begingroup$ Thanks for pointing this out. Is there some nice characterization of all 3d convex bodies that have constant diamter (or width) shadows? $\endgroup$
    – Nandakumar R
    Commented Aug 9, 2023 at 3:41
  • $\begingroup$ And what could be said about those convex solids which the ratio between diameter and least width is constant for all shadows? $\endgroup$
    – Nandakumar R
    Commented Aug 9, 2023 at 4:14

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