Powers of Chords for Convex Sets

The powers of chords S_n of planar convex sets are generally known only for n = 0, 1, and 3. They depend on the special shape of the convex set. Here integral formulas for the equilateral triangle, the square, and the ellipse are given. Furthermore, using area moments of the convex sets, S₅ and S₇ are derived in general form. For the 3-dimensional space S₆ and S₈ are given.