# Nonsymmetric conical upper density and $k$-porosity

## TL;DR

This paper investigates the distribution of Hausdorff measure in nonsymmetric cones and establishes an upper bound near $n-k$ for the dimension of $k$-porous sets, which have holes in multiple directions at small scales.

## Contribution

It introduces new bounds on Hausdorff dimension for $k$-porous sets based on measure distribution in nonsymmetric cones.

## Key findings

- Upper bound close to $n-k$ for Hausdorff dimension of $k$-porous sets
- Distribution analysis of Hausdorff measure in nonsymmetric cones
- Characterization of sets with holes in multiple directions

## Abstract

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in $\mathbb{R}^n$. As an application, we find an upper bound close to $n-k$ for the Hausdorff dimension of sets with large $k$-porosity. With $k$-porous sets we mean sets which have holes in $k$ different directions on every small scale.