The duality conjecture for metric entropy (Pietsch)

Theorem 1.1: for every $a,b\ge1$ there exist $n$ and an origin-symmetric convex body $K\subset\mathbb R^n$ such that, with $L$ the cube, $\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ)$. A further construction (per the Lean scope note) makes the dual-to-primal entropy ratio tend to zero. This disproves the two-sided dimension-free conjecture. It leaves intact the ellipsoid case, the convexified-packing duality, bounds with constants depending on the K-convexity constant, and Milman's logarithmic-loss comparison; no fixed-dimension claim is made.

Theorem 1.1: for every $a,b\ge1$ there exist $n$ and an origin-symmetric convex body $K\subset\mathbb R^n$ such that, with $L$ the cube, $\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ)$. A further construction (per the Lean scope note) makes the dual-to-primal entropy ratio tend to zero. This disproves the two-sided dimension-free conjecture. It leaves intact the ellipsoid case, the convexified-packing duality, bounds with constants depending on the K-convexity constant, and Milman's logarithmic-loss comparison; no fixed-dimension claim is made.