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We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension of ν and bounded above by a unique number , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.
We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various...
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