The size of the shadow boundary projection (Q1387106)

The paper contains the proof of the following Theorem. If a \(d\)-dimensional closed oriented \(C^\infty\)-manifold \(M\) with positive exterior curvature \(\leq k\) is embedded in a \((d+1)\)-dimensional Euclidean space, then the size \(L(M)\) of the shadow boundary projection is bounded by the size of the shadow boundary of the \(d\)-sphere \(S_k^d\) of radius \(1/k\) in the following way: \[ L(M)\leq \frac{V(M)}{V(S_k^d)} L(S_k^d), \] where \(V\) is the volume of the body.