关于两个对偶关系的注记
A remark on two duality relations
摘要：
我们注意到，一个简单的组合，两个已知的结果产生一个积极的答案，最多log（n）项，一个对偶猜想，可以追溯到Pietsch。特别地，我们证明了对于R^n中任意两个对称的凸体K，T，用N（K，T）表示T覆盖K所需的最小平移次数，有：N（K，T）<= N（T*，（Clog（n））^{-1} K*）^{Clog（n）loglog（n）}，其中K*，T* 分别是K，T的极体，C > 1是普适常数.作为推论，我们观察到Talagrand γ_p泛函的一个新的对偶结果（直到log（n）项）.
英文摘要：
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 R^n, denoting by N(K,T) the minimal number of translates of T needed to cover K, one has: N(K,T) <= N(T*,(C log(n))^{-1} K*)^{C log(n) loglog(n)}, where K*,T* 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 γ_p functionals.
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