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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1$\begingroup$ 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. $\endgroup$– Wlodek KuperbergCommented Aug 4, 2013 at 15:13
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2$\begingroup$ (one may also factorize the title as $($a question about$)^3$ 3-dimensional etc.) $\endgroup$– Pietro MajerCommented Aug 4, 2013 at 18:18
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$\begingroup$ @PietroMajer: This will make the title shorter, too. I have not thought about it... Will edit the title right away! $\endgroup$– Wlodek KuperbergCommented Aug 4, 2013 at 19:31
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1$\begingroup$ Was this another question from polymath? It's a very natural question. I'm amazed it wasn't asked before. meta.mathoverflow.net/questions/437/… $\endgroup$– David WhiteCommented Jan 27 at 1:36
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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.
answered Aug 4, 2013 at 16:05
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3$\begingroup$ Could you give a specific example for the case $n=3$, please? $\endgroup$ Commented Aug 4, 2013 at 16:48
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2$\begingroup$ 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. $\endgroup$ Commented Aug 4, 2013 at 16:54
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2$\begingroup$ @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. $\endgroup$ Commented Aug 4, 2013 at 21:15
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2$\begingroup$ Also related: there are non-spherical convex bodies such that the maximal cross sectional area in every direction is the same. See mathoverflow.net/questions/70391/… $\endgroup$ Commented Aug 4, 2013 at 21:36
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4$\begingroup$ 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: aimath.org/ARCC/workshops/sectionsconvex.html $\endgroup$ Commented Aug 4, 2013 at 22:07