# Variations on a problem of S. Mazur

In problem 76 of the Scottish Book Mazur asked

Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the length of the segment $op$ equals the area of the intersection of the body $K$ and the plane perpendicular to the segment and passing through the point $o$. Is the surface S convex?

Convex geometers will recognize this problem as a very elegant formulation of the question: given a convex body $K \subset \mathbb{R}^3$ with the origin in its interior, is the intersection body of $K$ also convex?

A celebrated theorem of Busemann---who apparently didn't know he was solving one of the problems in the Scottish Book---states that a sufficient condition for $S$ to be a convex surface is that $K$ be symmetric about the point $o$. This also holds for the $n$-dimensional version of the problem. Not to rob the theorem of its context, let me state it in a different form:

Theorem (H. Busemann). An $(n-1)$-dimensional disc lying on a hyperplane of an $n$-dimensional normed space has no more area (Hausdorff $(n-1)$-dimensional measure) than any competing $(n-1)$-dimensional hypersurface with the same boundary.

This formulation has the advantage of suggesting a natural generalization: Do $k$-dimensional flat discs in normed spaces minimize (Hausdorff) $k$-area? This is one of the Busemann-Petty problems and it was recently settled affirmatively for $k=2$ by D. Burago and S. Ivanov in this paper. However, it seems to me that Mazur's original formulation suggests another possible direction in which to generalize Busemann's theorem:

Question. Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the length of the segment $op$ equals the perimeter of the intersection of the body $K$ and the plane perpendicular to the segment and passing through the point $o$. Is the surface S convex?

Just like the original question, this one should be interpreted as "give general sufficient conditions on $K$ for $S$ to be convex". In higher dimensions we can play with other elementary mixed volumes besides area and perimeter.

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