Constancy results for special families of projections (Q2842343)

Let \(\{\Omega=V\times {\mathbb R}^l:V\in G(n-1,m-1)\}\) be the family of \(m\)-dimensional subspaces of \({\mathbb R}^n\) containing \(\{0\}\times {\mathbb R}^l\), and let \(\pi_{\Omega}: {\mathbb R}^n \rightarrow \Omega\) be the orthogonal projection onto \(\Omega\). The authors prove that the mapping \(V \mapsto \text{Dim}\;\pi_{\Omega}(B)\) is almost surely constant for any analytic set \(B \subset {\mathbb R}^n\), where \(\text{Dim}\) denotes either the Hausdorff or packing dimension.