Rigidity of sets of independent functions in symmetric spaces
We say that a symmetric function space $X$ has the $(IR)$ property whenever all sets of $N$ independent mean zero functions $f_1,\ldots,f_N\in X$, $\|f_k\|_X\ge 1$, are poorly approximated by any linear combinations of arbitrary $n$ functions, if $n$ is sufficienly smaller that $N$; namely, for some $γ=γ(X)>0$ we have $d_n(\{f_1,\ldots,f_N\},X)\ge γ$, $n\le γN$, where $d_n(K,X)$ is the Kolmogorov $n$-width of the set $K\subset X$. The spaces $X=L_p$ satisfy this property if and only if $1\le p\le2$ or $p=\infty$. The goal of this paper is to move from $L_p$ scale to a larger class of symmetric spaces. We obtain rather broad conditions, under which such a space $X$ has the $(IR)$ property and prove precise statements for particular scales of Lorentz $L_{p,q}$ spaces and Orlicz spaces.