The Definition of Convex Surface
Let $\Sigma$ be a $C^{\infty}$ compact surface in $R^3$.
(1)If the tangent space of every point lies the same side of $\Sigma$, we call $\Sigma$ convex surface.
(2)If the Guass Curvature $K>0$, we call $\Sigma$ ovaloid.
(3)If $\Sigma$ is homeomorphic to $S^2$, and the union of interior of $\Sigma$ and $\Sigma$ is convex, we call it a (3)-surface.
$\endgroup$ 13What is the relationship between convex surface and ovaloid? How is the situation in higher dimension?
In $R^n$, if $S$ is a compact convex set, then the boundary of $S$ is homeomorphic to a sphere. So is the definition (3) equal to the definition (1)?
1 Answer
$\begingroup$(1) and (3) are equivalent (this also holds in higher dimensions).
Suppose (1) holds. Each tangent plane divides the space into two half-spaces, one of which contains $\Sigma$. Take the intersection of all such halfspaces: it is a convex set containing $\Sigma$ and its interior. Show that it is equal to the union of $\Sigma$ and its interior. To prove that $\Sigma $ is homeomorphic to $S^2$, use radial projection from an interior point.
Suppose (3) holds. Convexity implies that for each $p$ there is a supporting plane: a plane that passes through $p$ and has the set on one side. Show that this plane is the tangent plane at $p$.
(2) implies the other properties (in 3 dimensions), but this is not trivial. The key words are Hadamard's ovaloid theorem
(1)-(3) do not imply (2); counterexamples are found in comments.
(1)-(3) imply $K\ge 0$. Indeed, suppose $K<0$ at some point $p$. Choose a system of coordinates so that $p$ is the origin and the tangent plane at $p$ is the $xy$-plane. Then the surface $\Sigma$ is locally the graph of a function $z=f(x,y)$ with a saddle point, because the Hessian determinant of $f$ is negative (see here). This contradicts (1).
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