Stability of the convex Pompeiu sets (Q1570190)

Let for some bounded set \(\Omega\) the function \(f=0\) be the only continuous function on the plane for which \[ \int_{\sigma(\Omega)} f(x) dx=0 \] for any rigid motion \(\sigma\) of the plane, then we say that a set \(\Omega \) is a Pompeiu set. The Pompeiu problem is to find all the bounded sets which are Pompeiu sets. The author proves that the Pompeiu convex domains are stable, i.e. if \(\Omega\) is a convex Pompeiu domain, then every convex domain close of \(\Omega \) is a Pompeiu set.