Generalized Mandelbrot rule for fractal sections

Mandelbrot’s rule for sections is generalized to apply to the Hentschel and Procaccia fractal dimension at arbitrary q and on arbitrary sections. It is shown that for almost all (n-m)-dimensional sections, ${\mathit{D}}^{\mathit{n}}$(q)=${\mathit{D}}^{\mathit{n}\mathrm{-}\mathit{m}}$(q)+m, where the ${\mathit{D}}^{\mathit{r}}$(q) are box-counting, Hentschel, and Procaccia generalized fractal dimensions of r-dimensional sections of homogeneous fractal point sets in ${\mathit{R}}^{\mathit{n}}$ and ${\mathit{D}}^{\mathit{n}\mathrm{-}\mathit{m}}$(q)>0. The rule applies for finite ‘‘thickness’’ sections as well as ‘‘true’’ sections and can be interpreted for inhomogenous fractal sets.