# Area of a surface confined by a sphere II

[A followup on two related posts: Area of a surface confined by a sphere

Area of a elliptic surface confined by a sphere

. Thanks to all the inputs so far.]

Let $$S$$ be a surface enclosed inside the unit sphere in $$R^3$$. If

1. every point of $$S$$ is elliptic and
2. there is a point $$p$$ inside the unit sphere so that every half-ray emanating from $$p$$ intersects $$S$$ at most once,

then must it be the case that $$\operatorname{Area}(S)\le \operatorname{Area}(S^2)$$?

• I do not think it is a good idea to ask almost identical question that many times. – Piotr Hajlasz Feb 16 at 5:07
• Are your assumptions equivalent with S being a subset of the boundary of a convex body? If so, then the inequality follows e. g. by monotonicity of mixed volumes. – Yoav Kallus Feb 17 at 17:09
• I think the answer is negative. (See below.) Thanks for your suggestion, that is indeed what I think I need. – Thomas Feb 18 at 1:29