Geodesic
$J$-closed finite collections of Hardy-type subspaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 82-92

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Several proofs of the following statement are given: if $X^0,\dots,X^n$ are BMO-regular lattices on the circle and $x\in X^0\cap\dots\cap X^n$, then the distances from $x$ to the Hardy-type subspaces $X^j_A$ are roughly attained at one and the same element of $\bigcap_jX^j_A$. 
@article{ZNSL_2012_401_a3,
     author = {P. Ivanishvili},
     title = {$J$-closed finite collections of {Hardy-type} subspaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {82--92},
     publisher = {mathdoc},
     volume = {401},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a3/}
}
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P. Ivanishvili. $J$-closed finite collections of Hardy-type subspaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 82-92. http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a3/