Some particular norm in the Sobolev space $H^{1}[a,b].$

This paper is a continuation of the recent paper of the author, where a certain reproducing kernel Hilbert space $X_{\mathcal{S}}$ was constructed. The norm in $X_{\mathcal{S}}$ is related to a certain generalized isoperimetric inequality in ${\mathbb R^2}$. In the present paper we give an alternative description of the space $X_{\mathcal{S}}$, which appears to be a Sobolev space $H^{1}[a,b]$ with some special norm.