Hausdorff dimension and non-degenerate families of projections

TL;DR

This paper investigates how the Hausdorff dimension of measures changes under parametrized families of projections, establishing optimal lower bounds for the dimension of almost all projections in various settings.

Contribution

It provides the best possible lower bounds for the Hausdorff dimension of projections in families with parameter spaces smaller than the Grassmann manifold.

Findings

Almost all projections preserve a significant portion of the original dimension.

Results apply to both orthogonal projections and smooth families of maps.

The bounds are optimal and extend previous understanding of dimension behavior under projections.

Abstract

We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from $n$-dimensional Euclidean space to $m$-dimensional one.