Smooth maps, null-sets for integralgeometric measure and analytic capacity (Q1073940)

A Borel subset E of the plane \(R^ 2\) has zero integralgeometric measure, \(I^ 1(E)=0\), if the orthogonal projections of E on almost all lines through the origin are of length zero. It is shown that a \(C^ 2\) diffeomorphism between two open subsets of \(R^ 2\) preserves sets of integralgeometric measure zero if and only if it maps every line segment onto a line segment. In particular, the condition \(I^ 1(E)=0\) is not conformally invariant. This disproves Vitushkin's conjecture according to which the compact null-sets for integralgeometric measure would be the same as the compact null-sets for analytic capacity.