2
$\begingroup$

Ref: Shadows and planar sections of polyhedra

By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that plane and the light at L=(0,0,+∞). C an be freely rotated as it hovers above the xy-plane. Are the following claims easy to prove/counter?

  1. If a convex 3D body C has all its shadows (orthogonal projected onto any plane) to be of equal area, then, C is a sphere.

  2. If a convex 3D body C has all its shadows to be of equal perimeter, then, C is a sphere.

Note added on 6th August 2023: This question has been answered in comments below.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ your question #1 seems to be a duplicate of mathoverflow.net/questions/138525/… $\endgroup$
    – Yoav Kallus
    Commented Jul 6, 2023 at 13:52
  • 1
    $\begingroup$ and @alvarezpaiva's comment on my answer to that question should give you a good hint for question #2, if you replace the square of the radial function by the support function $\endgroup$
    – Yoav Kallus
    Commented Jul 6, 2023 at 13:56
  • $\begingroup$ Thank you for those pointers. I guess, despite the repetition, this post adds a little bit of value in the second claim (if you think otherwise, do comment). A further claim has also been posted: mathoverflow.net/questions/450310/… $\endgroup$
    – Nandakumar R
    Commented Jul 6, 2023 at 19:38
  • $\begingroup$ you're welcome. Feel free to write up an answer to #2 if you think it adds value $\endgroup$
    – Yoav Kallus
    Commented Jul 7, 2023 at 1:51
  • 1
    $\begingroup$ As given in en.wikipedia.org/wiki/Girth_(geometry), the girth of a geometric object, in a certain direction, is the perimeter of its parallel projection in that direction (perimeter of the shadow). From Hilbert and Cohn-Vossen's 'Geometry and the Imagination', page 216-7, Minkowski has proved that all convex surfaces of constant girth (ie. all its shadows have constant perimeter), then the surface is one of constant width - not necessarily a sphere. This answers question 2. $\endgroup$
    – Nandakumar R
    Commented Aug 6, 2023 at 12:00

0

Reset to default

You must log in to answer this question.