Problem.Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\frac12\lambda(D)$?

(I.V.Protasov called such partitions *kaleidoscopic*).

Observe that for the $\ell_1$- or $\ell_\infty$-norms on the plane such partitions exist: just take a suitable chessboard coloring.

The problem can be reformulated in terms of convolutions: *Is there a measurable function $f:\mathbb R^2\to\{1,-1\}$ such that its convolution with the characteristic function $\chi_D$ of the unit disk $D$ is identically zero?*

(The problem was posed 08.11.2015 by T.Banakh and I.Protasov on page 19 of Volume 0 of the Lviv Scottish Book).