Convex generic submanifolds in an affine space (Q1356265)

A tangent hyperplane to a smooth submanifold \(M\) in the affine space \(\mathbb{R}^n\) is a support hyperplane if \(M\) lies entirely in one of the two closed half-spaces defined by this hyperplane. The manifold \(M\) is convex if, for every of its points, there is a support hyperplane passing through this point. A support hyperplane to the manifold \(M\) is nonsingular if it has only one point in common with \(M\) and this point is a nondegenerate critical point for every function obtained by the restriction to \(M\) of any nonzero linear function in \(\mathbb{R}^n\) which is equal to 0 in the hyperplane. The manifold \(M\) is strictly convex if, for every of its points, there is a nonsingular support hyperplane passing through this point. A strictly convex manifold is obviously convex. Generally speaking, the converse statement is not true. In this paper, we give a sketch of the proof of the following theorem. Any smooth closed convex \(k\)-dimensional generic submanifold in \(\mathbb{R}^n\), where \(k=1\) or \(n\leq 7\), is strictly convex. A detailed proof will be published in [Proc. Steklov Inst. Math. 209, 174-190 (1995); translations from Tr. Mat. Inst. Steklova 209, 200-219 (1995)].