Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the *closed* ball centered at 0 of radius 1.


I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a G\_\delta set, whose Minkowski's sum is not Borel. But can we have an example with one of them being a closed ball?



Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Idea from <http://arxiv.org/abs/1211.0430> (Example 2.4).


Take a Borel set $A' \subset [0,1]^2$ with the property that its projection to the first coordinate is not Borel. Now put this set on a cylinder in $\mathbb{R}^3$ and call it $A$. 
The set $$(A + B(0,1)) \cap (\mathbb{R}\times\{0\}^2)$$ (the Minkowski sum intersected with the axis of the cylinder) is the same as the non-Borel projection. Hence $A + B(0,1)$ is not Borel.


