*Accepted Paper*

**Inserted:** 22 oct 2013

**Last Updated:** 19 mar 2014

**Journal:** Proc. Amer. Math. Soc.

**Year:** 2013

**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 $\mathcal{C}$?

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