TAILIEUCHUNG - Đề tài " Duality of metric entropy "

Annals of Mathematics By S. Artstein, V. Milman, and S. J. Szarek For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is defined as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂. | Annals of Mathematics Duality of metric entropy By S. Artstein V. Milman and S. J. Szarek Annals of Mathematics 159 2004 1313-1328 Duality of metric entropy By S. Artstein V. Milman and S. J. SzAREK Abstract For two convex bodies K and T in Rra the covering number of K by T denoted N K T is defined as the minimal number of translates of T needed to cover K. Let us denote by K the polar body of K and by D the euclidean unit ball in Rra. We prove that the two functions of t N K tD and N D tK are equivalent in the appropriate sense uniformly over symmetric convex bodies K c Rra and over n E N. In particular this verifies the duality conjecture for entropy numbers of linear operators posed by Pietsch in 1972 in the central case when either the domain or the range of the operator is a Hilbert space. 1. Introduction For two convex bodies K and T in Rra the covering number of K by T denoted N K T is defined as the minimal number of translates of T needed to cover K N K T min N 3X1 .XN E Rra K c u Xi T . We denote by D the euclidean unit ball in Rra. In this paper we prove the following duality result for covering numbers. Theorem 1 Main Theorem . There exist two universal constants a and ft such that for any dimension n and any convex body K c Rra symmetric with respect to the origin 1 N D a-1K 1 N K D N D aK f where K u E Rra sup eK x Ù 1 is the polar body of K. This research was partially supported by grants from the US-Israel BSF all authors and the NSF . the third-named author . 1314 S. ARTSTEIN V. MILMAN AND S. J. SZAREK The best constant 3 that our approach yields is 3 2 e for any e 0 with a a e . Our theorem establishes a strong connection between the geometry of a set and its polar or equivalently between a normed space and its dual. Notice that since the theorem is true for any K we can actually infer that for any t 0 2 3-1 log N D a-1tK log N K tD 3 log N D atK . For definiteness above and in what follows all logarithms are to the base 2. The quantity log N

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