The author considers a sort of best approximation property in Banach space \(X\) as follows: Let \(W\) be a linear subspace of \(X\). A point \(y_0\in W\) is said to be a best approximation for \(x\in X\) if \[ \|x- y_0\|= d(x, W)= \inf\{\|x-y\|: y\in W\}. \] If each \(x\in X\) has at least one best approximation in \(W\), then \(W\) is called a proximal subspace of \(X\). If each \(x\in X\) has a unique best approximation in \(W\), then \(W\) is called a Chebyshev subspace of \(X\). For \(x\in X\), put \[ P_W(x)= \{y\in W:\|x-y\|= d(x, W)\}. \] It is clear that \(P_W(x)\) is a bounded, closed and convex subset of \(X\). A linear subspace \(W\) of a Banach space \(X\) is called quasi-Chebyshev if \(P_W(x)\) is a nonempty and compact set in \(X\) for every \(x\in X\). For a proximinal subspace of codimension one, he obtains the following result: Theorem. Let \(X\) be a Banach space and let \(W\) be a proximinal subspace of \(X\) with codimension one. Then the following are equivalent: (1) \(W\) is quasi-Chebyshev in \(X\). (2) Each sequence \(\{y_n\}_{n\geq 1}\) in \(X\) with \(\|y_n\|= 1\) and \(0\in P_W(y_n)\) \((n= 1,2,\dots)\) has a convergent subsequence. Also, for arbitrary proximinal subspace \(W\), he gets equivalent conditions for that \(W\) is quasi-Chebyshev in \(X\).