Smooth Maps, Null-Sets for Integralgeometric Measure and Analytic Capacity

Here IFI is the one-dimensional Lebesgue measure of a set F lying on a line, and p0 denotes the orthogonal projection onto the line Lo through the origin having normal (cos 0, sin 0). We call a map segmental if it maps every line segment onto a line segment. Note that there are nonaffine segmental diffeomorphisms; for example (x, y) -* (1/x, y/x). The converse of the above theorem also holds: If f is a segmental diffeomorphism, then I p0E I = 0 for almost all O e [0, ST) implies Ijp(JE) I = 0 for almost all 0 E [0, ST). In fact, an application of Fubini's theorem shows that IpoE I = 0 for almost all 0 if and only if almost all lines through almost any point of A do not meet E. This latter condition is easily seen to be preserved under segmental diffeomorphisms. We can restate the above remarks by saying that a C2 diffeomorphism preserves the null-sets for the integralgeometric (Favard) measure if and only if it is segmental. For a Borel set E C R2 the integralgeometric measure is given by