We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Ampére equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary.
"Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature." J. Differential Geom. 57 (2) 239 - 271, February, 2001. https://doi.org/10.4310/jdg/1090348111