14
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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 Kuperberg Aug 4 '13 at 15:13
  • 2
    $\begingroup$ (one may also factorize the title as $($a question about$)^3$ 3-dimensional etc.) $\endgroup$ – Pietro Majer Aug 4 '13 at 18:18
  • $\begingroup$ @PietroMajer: This will make the title shorter, too. I have not thought about it... Will edit the title right away! $\endgroup$ – Wlodek Kuperberg Aug 4 '13 at 19:31
15
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ Could you give a specific example for the case $n=3$, please? $\endgroup$ – John Bentin Aug 4 '13 at 16:48
  • 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. $\endgroup$ – Yoav Kallus Aug 4 '13 at 16:54
  • 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. $\endgroup$ – alvarezpaiva Aug 4 '13 at 21:15
  • 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/… $\endgroup$ – Yoav Kallus Aug 4 '13 at 21:36
  • 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 $\endgroup$ – Yoav Kallus Aug 4 '13 at 22:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.