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[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 $\operatorname{Area}(S)\le \operatorname{Area}(S^2)$?

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    $\begingroup$ I do not think it is a good idea to ask almost identical question that many times. $\endgroup$
    – Piotr Hajlasz
    Commented Feb 16, 2020 at 5:07
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    $\begingroup$ 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. $\endgroup$
    – Yoav Kallus
    Commented Feb 17, 2020 at 17:09
  • $\begingroup$ I think the answer is negative. (See below.) Thanks for your suggestion, that is indeed what I think I need. $\endgroup$
    – Thomas
    Commented Feb 18, 2020 at 1:29

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The two conditions in the question do not imply that the convex hull of S and p has S as part of its boundary. Moreover, I believe that the two conditions do not imply the desired conclusion. Here is a counterexample to the 1-D version of the question; it seems obvious how to extend it to the 2-D case.

The outer part of the red curve is almost as long as the circle, it makes a sharp turn to add a lot of lengths but without violating either condition.

In the 2-D case, make a surface with an outer part that is close to the confining sphere, then at a 'slit' the surface makes a sharp turn and adds a flat plate to its outer part.

p.s. I apologize for posting many similar questions.

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