ChapterZero

Craig Larson, a lecturer here at UH wrote an article that made it into Notices of the AMS on the current inadequacies in our recruitment of science and math teachers: the fact that they aren’t required to know the subject before they are allowed to teach it, and the money issues which cause those who would make the best teachers to find another field to work in. It’s a short article, and worth reading.

That plug aside, here’s an interesting and possibly elementary open problem from the SIAM Problemslist:

A sphere centered at the origin satisfies the following property:

where denotes Lebesgue measure and is a real function. That is, the volume of the intersection of and a scaled version of translated to one of its boundary points does not depend upon which boundary point is chosen; it is solely dependent upon the scale . This property can be shown to hold for ellipsoids by appropriately scaling the coordinates of a sphere.

Are there other domains centered at the origin which satisfy the above equation?

I changed the question somewhat, since I don’t understand what the terms homothety or homothetic center refer to, but it retains the spirit of the original.