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For a given domain $ D $ in the extended complex plane $\overline {\mathbb C} $ with an accessible boundary point $ z_0\in\partial D $ and for a subset $ E\subset {D}, $ relatively closed wrt $ D, $ we define the relative capacity $\textrm {rel cap}{} E $ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $ r (D\setminus E, z)/r (D, z) $ when $ z $ approaches the point $ z_0. $ Here $ r (G, z) $ denotes the inner radius at $ z $ of the connected component of the set $ G $ containing the point $ z. $ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $ D $ is the upper half plane and $ z_0=\infty $, the capacity $\textrm {rel cap}{} E $ coincides with the well-known half-plane capacity $\textrm {hcap}{} E. $ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization …