I've just started reading about convex surfaces and there are a few things which are breaking my intuition. According to this page: "Minkowski proved the existence of a closed convex surface with given Gaussian curvature."
Since there cannot be a closed surface embedded in $\mathbb{R}^3$ which has negative curvature everywhere, the result must be referring to particular points on the surface. Even so, wouldn't the saddle shape that the surface would take on near the point with negative curvature violate convexity? I have difficulty seeing how any convex surface could have a point with negative curvature, let alone a closed one.
According to this answer, the boundary of a convex body always has negative curvature when curvature is defined. This makes sense, but if a general convex surface is a connected open set of the boundary of a convex body then doesn't this result about non-negative curvature also apply to it?
