Slices and $m$-Lelong numbers of $m$-subharmonic functions

We investigate slicing properties of $m$-subharmonic functions in product domains $\mathrm{\Omega }={\mathrm{\Omega }}^{\mathrm{\prime }}\times {\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}\subset {\mathbb{C}}^n={\mathbb{C}}^p\times {\mathbb{C}}^{n-p}$, where $p,m,n$ are integers satisfying $1\le p\le m-1<n-1$.\ Given an $m$-subharmonic function $v$ on $\mathrm{\Omega }$, we prove the existence of a pluripolar subset $E\subset {\mathrm{\Omega }}^{\mathrm{\prime }}$ such that, for every $x^{\mathrm{\prime }}\in {\mathrm{\Omega }}^{\mathrm{\prime }}\setminus E$, the slice $v_{\mathrm{\mid }\{x^{\mathrm{\prime }}\}\times {\mathbb{C}}^{n-p}}$ is well defined and $(m-q_{m,p})$-subharmonic on ${\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}$, where $q_{m,p}$ denotes the smallest integer greater than or equal to $\frac{mp}{n}$.\ Moreover, we show that, outside a negligible subset of ${\mathrm{\Omega }}^{\mathrm{\prime }}$, the $m$-Lelong number of $v$ at $(x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }})$ coincides, up to a multiplicative constant, with the $(m-q_{m,p})$-Lelong number of the slice $v_{\mathrm{\mid }\{x^{\mathrm{\prime }}\}\times {\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}}$ at $x^{\mathrm{\prime }\mathrm{\prime }}$.