On locally convex hypersurfaces with boundary

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The affine Plateau problem (2004, background): “This property was crucial for our resolution of the Plateau problem for prescribed constant Gauss curvature [22] and also plays a key role in the following existence proof of maximizers to the affine Plateau problem.”
Affine complete locally convex hypersurfaces (2002, background): “Remark 1. Theorem 3.1 has been extended to nonsmooth hypersurfaces in [ 14 ], where we also found other geometric characterizations of locally convex hypersurfaces with boundary.”
Interior curvature bounds for a class of curvature equations (2004, methodology): “The second observation is that Lemma 5.1 holds not just for M0, but also for a family of locally convex hypersurfaces, with uniform r and . Indeed, by extending M0 to a larger locally convex hypersurface M1 such that @M0 lies in the interior of M1 (see [ TW ]), and applying Lemma 5.1 to M1, we see t”
Locally convex hypersurfaces of constant curvature with boundary (2004, methodology): “This method is based on the solvability of the problem in the non-parametric setting (the Dirichlet problem) and an important uniform local graph representation property of hypersurfaces in A[] (Theorem 3.1) which was first proved in [12], and independently in [ 24 ] in somewhat less generality.”