Abstract
Let S n {S^n} be the Euclidean sphere of dimension n. Let p and q be antipodal points on S n {S^n} , and, for nonnegative h, let C ( p , h ) , C ( q , h ) 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 {S^n} with mean curvature bounded by h must have a point in the ’tropical’ region bounded by C ( p , h ) C(p,h) and C ( q , h ) C(q,h) .
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