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Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all directions are of the same area?

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Wlodek Kuperberg
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    In 3-d the average projected area of a convex solid is 1/4 the surface area, as Cauchy showed in the 19th century (see arXiv:1109.0595v4). Thus the surface area of the body in question is determined by the area of its shadow. – Wlodek Kuperberg Aug 04 '13 at 15:13
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    (one may also factorize the title as $($a question about$)^3$ 3-dimensional etc.) – Pietro Majer Aug 04 '13 at 18:18
  • @PietroMajer: This will make the title shorter, too. I have not thought about it... Will edit the title right away! – Wlodek Kuperberg Aug 04 '13 at 19:31
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    Was this another question from polymath? It's a very natural question. I'm amazed it wasn't asked before. https://meta.mathoverflow.net/questions/437/how-to-treat-polymath-questions – David White Jan 27 '24 at 01:36

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A convex body $K\subset\mathbb{R}^n$ all of whose $(n-1)$-dimensional projections have the same $n-1$-content is known as a body of constant brightness, by analogy with bodies of constant width. The theory is very similar to that of bodies of constant width. The surface area measure $dS_K(\mathbf{x})$ takes the place of the support function $h_K$. The brightness in the direction $\mathbf{u}$ is given by $V(K_\mathbf{u}) = \tfrac{1}{2}\int |\langle\mathbf{x},\mathbf{u}\rangle| dS_K(\mathbf{x})$. If we expand $S_K$ in spherical harmonics $\sum_{l,m}s_{l,m}Y_{l,m}$, we get that $V(K_\mathbf{u}) = \sum_{l,m} c_l s_{l,m} Y_{l,m}(\mathbf{u})$, and $c_l=0$ whenever $l$ is odd. Therefore, $K$ has constant brightness if and only if $\frac{S_K(U)+S_K(-U)}{|U|}$ is constant over all Borel sets $U\subset S^{n-1}$ (that is, apart from a constant term, $S_K$ is antisymmetric). From the existence theorem for the Minkowski problem, we can easily construct examples.

Yoav Kallus
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    Could you give a specific example for the case $n=3$, please? – John Bentin Aug 04 '13 at 16:48
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    Let $S_K(x,y,z) = 1 + a P_3(z)$, where $P_3(z)=(5z^3-3z)/2$ is the Legendre Polynomial, and $a$ is small enough so that $S_K$ is non-negative. By the existence theorem for the Minkowski problem, this is the surface area measure of some convex body. – Yoav Kallus Aug 04 '13 at 16:54
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    Among bodies of revolution with constant brightness, the Blaschke-Fiery body is of minimal volume. That's the body whose surface area measure is constant for $0<z<a_n$ or $z<-a_n$ and zero otherwise, where $a_n = 1/2$ in $n=3$. See P. Gronchi, "Bodies of constant brightness", Archiv der Mathematik 70, pp. 489-498, 1998. – Yoav Kallus Aug 04 '13 at 17:48
  • That is, constant density relative to the Lebesgue measure on the sphere. – Yoav Kallus Aug 04 '13 at 17:49
  • @YoavKallus: Many thanks! And what about the problem of a convex body with a point in the interior through which cross-sections with all planes have equal areas - does polar duality take care of that? Or maybe it's not so simple... – Wlodek Kuperberg Aug 04 '13 at 19:38
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    @WlodzimierzKuperberg: if you consider the radial function of your convex body (with center at the special point), then using polar coordinates you'll see that your condition translates into a statement for the spherical Radon transform of the square of the radial function. If your body is symmetric about the point, then the body can only be a sphere, but if not you can have lots of examples. – alvarezpaiva Aug 04 '13 at 21:15
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    Also related: there are non-spherical convex bodies such that the maximal cross sectional area in every direction is the same. See http://mathoverflow.net/questions/70391/convex-bodies-with-constant-maximal-section-function-in-odd-dimensions – Yoav Kallus Aug 04 '13 at 21:36
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    Incidentally, I am writing this from a hotel in Palo Alto where I am staying for the duration of a workshop in the American Institute of Mathematics on "Sections of convex bodies" starting tomorrow: http://www.aimath.org/ARCC/workshops/sectionsconvex.html – Yoav Kallus Aug 04 '13 at 22:07