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We characterize submanifolds of Euclidean space which lie on closed hypersurfaces of positive curvature, and develop some applications of this result for boundary value problems via Monge-Ampere equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary. The main result of this dissertation states that every smooth compact submanifold M of Euclidean space lies embedded in a smooth closed hypersurface of positive curvature if, and only if, M is strictly convex, i.e., through every point of M there passes a hyperplane, with contact of order one, with respect to which M lies strictly on one side. As applications of this result we show: (1) Every smooth closed strictly convex submanifold of codimension two bounds a smooth hypersurface of constant positive curvature. (2) Let M be a closed strictly convex submanifold of codimension 2; then, if M is usdC\sp{3,1}usd, the two hypersurfaces making up the boundary of the convex hull of M are each usdC\sp{1,1}usd; this result is optimal. (3) Every polytope P may be approximated arbitrarily closely by a closed hypersurface of nonnegative curvature which coincides with the boundary of P everywhere outside any given open neighborhood of the singular points. (4) Let M be a compact connected hypersurface of positive curvature in Euclidean n-space, usdn \ge 3usd, then M is strictly convex, if, and only if, each boundary component of M lies strictly on one side of the tangent hyperplanes of M at that component. Furthermore, we discuss some applications for self-linking number of space curves, and umbilic points of ovaloids.
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