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Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function $$L\longmapsto -\log(\hbox{packing density of }L)/ \log(\hbox{covering density of }L)$$ is bounded and bounded away from $0$ if restricted to all lattices of a given dimension $d$.

What are the local minima and maxima (if there are any) of this function?

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