Let us call a *cap* the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this intersection is contained in $K\cap H$.


**Question.** Is it true that a cap equipped with the intrinsic metric is an Alexandrov space (with boundary) with non-negative curvature.


**Remark.** In the case when $K$ has smooth boundary and $H$ intersects $K$ transversally, the question is equivalent to asking whether the second fundamental form of the boundary of the cap is non-negative.



Is a cap an Alexandrov space?
Note that the union of $K\cap H$ with its reflection is a convex set.
Therefore, its surface $\Sigma$ is an Alexandrov space.
Your space is a quotient $\Sigma/\mathbb{Z}\_2$ by isometric involution.
Therefore, it is an Alexandrov space as well.


(There is a closely related problem *Convex hat*, page 21 in my [PIGTIKAL](https://arxiv.org/abs/0906.0290).)


