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For example, contract a hemisphere to a segment by contracting each radial circle to a point. The resulting space is a bouquet of a sphere and a circle. Place the segment to the sphere by an injective map. This can be made smooth.
Take a point with the highest $z$-coordinate. If it is on the boundary, take one with the lowest $z$-coordinate. Since the entire boundary is on the same level, either a highest or lowest point is in the interior.
@Anton, I don't get it. In any manifold, take a point $p$ and a nearby small ball $B$, and connect $p$ to all points of $B$ by geodesic segments. I believe the resulting set is convex if everything is small enough, yet the conical part of the boundary is filled by geodesics.
Judging by Google search results, self-dual cones were studied quite a bit. Since self-dual bodies in $R^d$ are precisely compact hyperplane sections of self-dual cones in $R^{d+1}$, it should be possible to translate many results from one context to the other.
@Misha, what about Helly's theorem? It boils down to the following: if the $(d+1)$-simplex is mapped to a space of dimension $d$, then the images of $d$-faces have a common point. This looks like something standard, but my knowledge of this stuff is too limited.
I'm afraid this function may fail to be $C^\infty$. Although $dist(\cdot,y)$ is smooth for a.e. $y$, its higher order derivatives may be large if $y$ approaches a singular point. Then, when computing the respective derivative of $\tilde d_p$, you may get a non-summable integral.
If $R$ has constant coordinates, then yes. Commuting with a vector field is the same thing as commuting with its flow. And since the orbit is dense, translations by $R$ can approximate any translation. Hence the operator commutes with translations (but not with the full isometry group, which includes other connected components).
