Uniform convexity and the distribution of the norm for a Gaussian measure

✍ Scribed by WanSoo Rhee; Michel Talagrand

• Publisher: Springer
• Year: 1986
• Tongue: English
• Weight: 371 KB
• Volume: 71
• Category: Article
• ISSN: 1432-2064
• DOI: 10.1007/bf00366272

✦ Synopsis

We show that if a Banach space E has a norm [I'll such that the modulus of uniform convexity is bounded below by a power function, then for each Gaussian measure # on E the distribition of the norm for # has a bounded density with respect to Lebesgue measure. This result is optimum in the following sense:

If (an) is an arbitrary sequence with a,--*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 ~(e)> en for e > a n, and a Gaussian measure # on E such that the distribution of the norm for # does not have a bounded density with respect to Lebesgue measure.