Avoiding \sigma-porous sets in Hilbert spaces

TL;DR

This paper proves that -porous sets in Hilbert spaces have measure zero on typical smooth curves, but shows that this property does not hold for all porous sets, especially power- porous sets.

Contribution

It provides a constructive proof for measure-zero properties of -porous sets on typical curves and highlights limitations for other porous sets.

Findings

-porous sets have Lebesgue measure zero on typical $C^{1}$ curves

Power- porous sets can intersect many $C^{1}$ curves in positive measure

The measure-zero property does not extend to all forms of porosity

Abstract

We give a constructive proof that any $\sigma$-porous subset of a Hilbert space has Lebesgue measure zero on typical $C^{1}$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-$p$ porous sets may meet many $C^{1}$ curves in positive measure.