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首页> 外文期刊>Glasgow Mathematical Journal >DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES
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DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

机译：在Banach空间中嵌入局部有限度量空间的有限确定结果中的失真

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Given a Banach space X and a real number alpha = 1, we write: (1) D(X) = alpha if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions = C, the space A itself admits a bilipschitz embedding into X with distortion = alpha . C; (2) D(X) = alpha(+) if, for every epsilon 0, the condition D(X) = alpha + epsilon holds, while D(X) = alpha does not; (3) D(X) = alpha(+) if D(X) = alpha(+) or D(X) = alpha. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((circle plus(infinity)(n= 1) X-n)(p)) = 1(+) for every nested family of finite-dimensional Banach spaces {X-n}(n=1)(infinity) and every 1 = p = 8 infinity. (2) D((circle plus 8(n=1)(infinity)l(infinity)(n) )(p)) = 1(+) for 1 p infinity. (3) D(X) = 4(+) for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier-Lancien result (2008).
机译：给定Banach空间x和实数alpha＆ = 1，我们写入：（1）d（x）＆ = alpha if，对于任何本地有限的公制空间a，所有有限子集都承认bilipschitz嵌入x扭曲＆ = c，空间a本身承认嵌入到具有失真x的x中的bilipschitz。 C; （2）D（x）= alpha（+）如果，对于每个epsilon＆ 0，条件d（x）＆ = alpha + epsilon保持，而d（x）<= alpha没有; （3）D（x）＆ = alpha（+）如果d（x）= alpha（+）或d（x）＆ = alpha。众所周知，D（x）由通用常数界定，但是这种常数的可用估计相当大。在这项工作中已经证明了以下结果：（1）D（（圆加（Infinity）（n = 1）xn）（p））＆ = 1（+）用于每个嵌套的有限维巴赫空间{ xn}（n = 1）（无穷大）和每个1＆ = p＆ = 8无穷大。 （2）D（（圆加8（n = 1）（无穷大）L（无穷大）（n））（p））= 1（+）1＆ P＆无限。 （3）D（x）＆ = 4（+）每个Banach空间x，没有非活动型。声明（3）是加强Baudier-Lancien结果（2008）。

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