The purpose of the paper is to characterize the dimension of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{ m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{ m+1}$. One should compare it to the result of Dranishnikov \cite{ Dr1} who characterized the dimension of the Higson corona $\nu(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{ n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{ n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{ n+1}$). \par In \cite{ Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(\nu_L(X))\leq n$ if and only if $\R^{ n+1}$ is an absolute extensor of $X$ in the category $\tilde\AAA$. \par As an application we reprove the following result of Dranishnikov and Smith \cite{ DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension $\asdim_{ AN}(X)$. If $X$ is cocompact and connected, then $\asdim_{ AN}(X)$ equals the dimension of the sublinear Higson corona $\nu_L(X)$ of $X$.
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