Summary
We show that if a Banach space E has a norm ‖·‖ such that the modulus of uniform convexity is bounded below by a power function, then for each Gaussian measure μ on E the distribition of the norm for μ has a bounded density with respect to Lebesgue measure. This result is optimum in the following sense:
If (a n) is an arbitrary sequence with a n→0, there exists a uniformly convex norm N(·) on the standard Hilbert space, equivalent to the usual norm such that the modulus of convexity of this norm satisfies \(\alpha (\varepsilon ) \geqq \varepsilon ^{n} {\text{ for }}\varepsilon \geqq a_n \), and a Gaussian measure μ on E such that the distribution of the norm for μ does not have a bounded density with respect to Lebesgue measure.
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Rhee, W., Talagrand, M. Uniform convexity and the distribution of the norm for a Gaussian measure. Probab. Th. Rel. Fields 71, 59–67 (1986). https://doi.org/10.1007/BF00366272
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DOI: https://doi.org/10.1007/BF00366272