On Burdick's symmetry problem. (Q1856516)

Let \(P\) be a probability measure on the real line \(R\) such that each of the product measures \(P^{\otimes n}\) assigns the value \(1/2\) to every half space in \(\mathbb R^n\) having the origin as a boundary point. Then \(P\) is symmetric. Example: a strictly stable law on \(R\) is symmetric iff it has median zero. The treated symmetry problem is related to the problem of characterizing the distribution of \(X_1\) by the distribution of \((X_2+X_1,\dots,X_n+X_1)\), with \(X_1,\dots,X_n\) being iid random variable \(s\).