Small ball probability estimates, ψ2-behavior and the hyperplane conjecture
Small ball probability estimates, ψ2-behavior and the hyperplane conjecture
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DOI：
10.1016/j.jfa.2009.06.038
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发表时间：
2010-03
影响因子：
1.7
通讯作者：
N. Dafnis;G. Paouris
中科院分区：
文献类型：
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作者：
N. Dafnis;G. Paouris
我们介绍了一种求各向同性常数上界的方法。我们证明了超平面猜想的一个正解等价于各向同性凸体上欧几里德范数的一些非常强的小概率估计。作为我们的方法的结果，我们获得了J. Bourgain的结果的另一种证明，即每个ψ -体都有有界各向同性常数，具有稍微更好的估计：如果K是rn中的对称凸体，对于每个θ∈Sn - 1和每个q∈Sn - 2，都有‖<⋅,θ >‖q≤β‖<⋅,θ >‖2，则LK≤Cβlogβ，其中C>0是绝对常数。
We introduce a method which leads to upper bounds for the isotropic constant. We prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the Euclidean norm on isotropic convex bodies. As a consequence of our method, we obtain an alternative proof of the result of J. Bourgain that every ψ2-body has bounded isotropic constant, with a slightly better estimate: If K is a symmetric convex body in Rnsuch that ‖〈⋅,θ〉‖q⩽β‖〈⋅,θ〉‖2for every θ∈Sn−1and every q⩾2, then LK⩽Cβlogβ, where C>0 is an absolute constant.