On the building dimension of closed cones and Almgren's stratification principle

TL;DR

This paper disproves a conjecture about the equality of two dimension notions for closed cones and addresses a question related to Almgren's stratification, providing negative answers and new insights into geometric measure theory.

Contribution

It demonstrates that the conjectured equality of two cone dimensions does not hold and answers negatively a question about the dimension bounds of sets based on their blow-ups.

Findings

Disproved a conjecture on the equality of two cone dimensions.

Provided a negative answer to a question on dimension bounds for sets with cone blow-ups.

Enhanced understanding of Almgren's stratification and geometric measure theory.

Abstract

In this paper we disprove a conjecture stated in [4] on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family $\mathcal{C}$ of closed cones and a set $S$ such that every blow-up of $S$ at every point $x\in S$ is contained in some element of $\mathcal{C}$, is it true that the dimension of $S$ is smaller than or equal to the largest dimension of a vector space contained is some element of $\mathcal{C}$?