Maxima position of the PDF of the parameter y_n and the normalized mass surface density X_n for spheres (top) and cylinders (bottom) as function of the power n (dotted curves). Black lines correspond to the linear PDF and the gray lines to logarithmic PDF. The solid curves are polynomial fits to the accurate position values. The values for isothermal self-gravitating spheres (n = 2 and n = 3) and cylinders (n = 4) are shown as small diamonds. The asymptotic behavior of [X_n] _max = 1/(1 − n) for n → 0 for cylinders is shown as dashed line. Asymptotes described in Appendix C.1.1 are shown as dashed lines.