The \(L^p(\mu)\)-Christoffel function is defined as \(\lambda_{n,p}(\mu, \zeta ) := \inf\{\Vert Q\Vert^p_p:\;Q\in\Pi_n,\;Q(\zeta)=1\}\), \(1\le p<\infty\), where \(\mu\) is a finite positive Borel measure on the unit circle \(\mathbb{T}\) with infinitely many points in its support and \(\zeta\) is a complex number. For \(p\in[1,\infty]\), denote by \(q\in[0,\infty]\) the conjugate exponent of \(p\). Let \(V_{n,q} (\mu, \zeta) := \{ f\in L^q(\mu):\;\int Q \overline{f} d\mu=Q(\zeta),\;\mbox{for every}\;Q\in\Pi_n\}\), \(p\in[1,\infty]\). Let \(\Omega_{n,q}(\mu,\zeta)=\inf\{\Vert f\Vert_q^p;\;f\in V_{n,q}(\mu,\zeta)\}\), \(1<q\le\infty\) and define \(\Omega_{n,1}\) as the infimum of the norm \(\Vert\cdot\Vert_1\). It is obtained that \(\Omega_{n,q}(\mu,\zeta)=(\lambda_{n,p}(\mu, \zeta ))^{-1}\), if \(1\le p<\infty\) and \(|\zeta|=1\). Also other links are established between \(\Omega_{n,q}(\mu,\zeta)\) and certain extremal problems built with the aid of the Hardy spaces \(H^p\).