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Construct pairs of $n$-dimensional convex bodies with given ratios ($p$) of volumes

Given a dimension $n$ and a number $p \in (0,1)$, to what extent is it possible (in what cases) to construct a convex set $A$--not a hypersphere--and a "snugly" inscribed (InscribedFigure) ...
Paul B. Slater's user avatar