Rate of Convergence Estimates for Random Polarizations on R^d

Details Rate of Convergence Estimates for Random Polarizations on R^d Rate of Convergence Estimates for Random Polarizations on R^d

2012-03-26

arxiv.org/abs/1203.5760v4

Mathematics

Functional Analysis

Scientific paper

It is shown in arXiv:1104.4103 that the expected $L^1$ distance between $f^*$ and $n$ random polarizations of a function $f$ - with support in a ball of radius $L$ - is bounded above by $2dm(B_{2L})||f||_{infty}n^{-1}$. Here it is shown that the expected distance is bounded below by $(1-c_d)^n||f-f^*||_{L^1}$ with $c_d<2^{-1}$ for all $d$ and, if the boundary of ${f>t}$ has measure zero for almost every $t$, then the expected distance equals $n^{-1}c_n(f)$ with $limsup_{n->infty}c_n(f)<= L2^{d+1}||nabla f^*||_{L^1}$. The paper concludes with the construction of a function $f$ whose distance from $f^*$ after $n$ random polarizations behaves asymptotically like $n^{-1}L\beta_d||nabla f^*||_{L^1}$ for some constant $\beta_d$ that only depends on $d$.

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