Mean square inequalities for chords of convex sets

This paper is concerned with establishing lower bounds for the integrals of the square of the lengths of area and perimeter bisecting chords of planar convex sets. The results obtained provide verification of two recent conjectures of Lutwak. When combined with the known upper bounds for these integrals they yield the classical isoperimetric inequality. The main proof technique involves estimation of the winding numbers of the locus of the midpoints of the chords concerned.