Constancy results for special families of projections

TL;DR

This paper proves that for a specific family of projections onto m-dimensional subspaces containing a fixed subspace, the dimension of the projected set is almost surely constant for analytic sets.

Contribution

It establishes the constancy of the dimension of projections for a particular family of subspaces, extending understanding of projection behaviors in geometric measure theory.

Findings

Dimension of projections is almost surely constant for analytic sets.

The result applies to both Hausdorff and packing dimensions.

The family of subspaces considered contains a fixed subspace and varies in a specific manner.

Abstract

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