The Gaussian inequality for multicomponent rotators
β Scribed by J. Bricmont
- Publisher
- Springer
- Year
- 1977
- Tongue
- English
- Weight
- 499 KB
- Volume
- 17
- Category
- Article
- ISSN
- 0022-4715
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π SIMILAR VOLUMES
Let + be a Gaussian measure (say, on R n ) and let K, L R n be such that K is convex, L is a ``layer'' (i.e., L=[x: a (x, u) b] for some a, b # R and u # R n ), and the centers of mass (with respect to +) of K and L coincide. Then +(K & L) +(K) } +(L). This is motivated by the well-known ``positive
Let g be a smooth function on R n with values in [0, 1]. Using the isoperimetric property of the Gaussian measure, it is proved that ,(8 &1 (Eg))&E,(8 &1 ( g)) E |{g|. Conversely, this inequality implies the isoperimetric property of the Gaussian measure.
After the completion of this work, Richard Hamilton proved that the diameter of any compact convex hypersurface in Euclidean space is bounded above in terms of its enclosed volume and an upper bound for its entropy.
A converse PoincarΓ e-type inequality is obtained within the class of smooth convex functions for the Gaussian distribution.