Multifractal geometry of slices of measures (Q2039400)

Let $m,n$ be integers with $0<m<n$. Consider a Borel probability measure $\mu$ in $\mathbb{R}^n$ with compact support. Given $q\in \mathbb{R}$, one can define the $(\mu,q)$-multifractal Hausdorff dimension of $\mu$, $b_\mu(q)$, as well as multifractal packing and pre-packing dimensions, which coincide with the usual dimensions when $q=0$. Note that for $q>1$, multifractal dimensions are non-positive. In [Period. Math. Hung. 37, No. 1--3, 81--99 (1998; Zbl 0944.28007)], \textit{L. Olsen} proved an estimate on the multifractal Hausdorff dimensions of slices by $n-m$ planes of a measure for $0\leq q<1$, under some technical assumptions: for almost every affine $n-m$ plane $V$, there holds \[ b_{\mu_V}(q) \leq b_\mu(q) -m(1-q). \] In the paper under review, the author proves similar results for the multifractal packing and pre-packing dimensions of slices for $1<q\leq 2$. He mentions that the result for the pre-packing dimension is contained in the paper of \textit{K. J. Falconer} and \textit{T. C. O'Neil} [Math. Nachr. 204, 61--82 (1999; Zbl 1040.28011)] under the assumption that the measure and its slices are doubling.