By A.M. Fink

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Na/ with Ei given by (3). We also have (with jj jj denoting the L2 -norm in jjf jj Ä jj ı Gjj C jjujj Ä . na/ C C /2 We are now going to use the tensor power trick. w; : : : ; w/ 2 e . z /; j D1;:::;m 58 Z. fGe;e w < ag/ D . nm; a/ . 1 : t u Theorem 2 follows immediately from Theorem 1 and the following result by approximation. Proposition 3. Assume that Cn . w//: (8) Proof. w/, G WD G ;w , we may assume that w D 0. By the results of Lempert [15] there exists a diffeomorphism ˆ W IN ! N such that for v 2 @I the mapping 3 7!

This way we would get a certain counterpart of logarithmic capacity in higher dimensions. Using Lempert’s theory [15, 16] one can A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality 55 check what happens with this limit for smooth and strongly convex domains, see Proposition 3 below. This way we get the following bound: Theorem 2. Let be a convex domain in Cn . 0/ D wg is the Kobayashi indicatrix (here denotes the unit disc). One can use Theorem 2 to simplify Nazarov’s approach [17] to the Bourgain-Milman inequality [6].

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