A Remark on Two Duality Relations

Abstract.

We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two symmetric convex bodies K, T in ${\user2{\mathbb{R}}}^{n}$, denoting by N(K, T) the minimal number of translates of T needed to cover K, one has:

$$ N(K,T) \leq N(T^{ \circ } ,(C\log (1 + n))^{{ - 1}} K^{ \circ } )^{{C\log (1 + n)\log \log (2 + n)}} $$

, where $K^{ \circ } ,T^{ \circ }$ are the polar bodies to K, T, respectively, and C  ≥ 1 is a universal constant. As a corollary, we observe a new duality result (up to log(n) terms) for Talagrand’s $\gamma _{p}$ functionals.