Position of compact hypersurfaces of the $n$-sphere

Abstract:

Let ${S^n}$ be the Euclidean sphere of dimension n. Let p and q be antipodal points on ${S^n}$, and, for nonnegative h, let $C(p,h),\;C(q,h)$ be the hyperspheres of constant mean curvature h centered at p and q, respectively. Then any closed hypersurface in ${S^n}$ with mean curvature bounded by h must have a point in the ’tropical’ region bounded by $C(p,h)$ and $C(q,h)$.