Defining a metric in a linear space by means of a family of subsets

Necessary and sufficient conditions are given on a family $\{A_r\}_{r>0}$ of subsets of a real linear space $X$ under which $\inf\{r>0:x\in A_r\}$ is a quasinorm [1] on X. It is shown that for any symmetric (about zero) closed set $A$ in a normed space $X$ containing the ball $\{x\in X:\|x\|\le1\}$ there exists a quasinorm $|\cdot|$ on $X$ such that $A=\{x\in X:\|x\|\le1\}$. Examples are constructed of linear metric spaces in which there exists a Chebyshev line which is not an approximately compact set.