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A classical theorem (Marstrand 1954) asserts that, given any Borel subset $X\subset\mathbb R^d$, the obvious inequality of the Hausdorff dimensions: $\dim(\pi(X))\leq \min(k,\dim(X))$ is in fact an equality for almost all orthogonal projections $\pi:\mathbb R^d\to\mathbb R^k$. As is often the case it is usually very dificult to prove equality for a given projection.