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Let $X$ be a metric space equipped with a doubling measure. We considerweights $w(x)=\operatorname{dist}(x,E)^{-\alpha}$, where $E$ is a closed set in$X$ and $\alpha\in\mathbb R$. We establish sharp conditions, based on theAssouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$classes of weights, $1\le p<\infty$. With the help of general $A_p$-weightedembedding results, we then prove (global) Hardy-Sobolev inequalities and alsofractional versions of such inequalities in the setting of metric spaces.

Muckenhoupt $A_p$-properties of distance functions and applications to Hardy-Sobolev -type inequalities