An area estimate for slices of strictly convex hypersurfaces (Q800643)

The following theorem is proved. Let D be a compact, strictly convex hypersurface (with positive curvatures) in \({\mathbb{R}}^ n\) (\(n\geq 3)\) of class \(C^ 2\) and let M be the part of D between two parallel hyperplanes distance \(\epsilon\) apart. For \(\epsilon\) sufficiently small, the area of M is bounded from above by \(C\epsilon\), where C is a constant depending only on D. The proof proceeds by direct computation and estimation.