On the building dimension of closed cones and almgren’s stratification principle

In this paper we disprove a conjecture stated in Stratification of minimal surfaces mean curvature flows and harmonic maps by Brian White on the equality of two notions of dimension for closed cones Moreover we answer in the negative the following question raised in the same paper Given a compact family C of closed cones and a set S such that every blow-up of S at every point x ∈ S is contained in some element of 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 C?