Summary: A new test is proposed for the hypothesis of uniformity on bi-dimensional supports. The procedure is an adaptation of the “distance to boundary test” (DB test) proposed in [J. R. Berrendero et al., Can. J. Stat. 34, No. 4, 693–707 (2006; Zbl 1115.62046)]. This new version of the DB test, called DBU test, allows us (as a novel, interesting feature) to deal with the case where the support \(S\) of the underlying distribution is unknown. This means that \(S\) is not specified in the null hypothesis so that, in fact, we test the null hypothesis that the underlying distribution is uniform on some support \(S\) belonging to a given class \(\mathcal C\). We pay special attention to the case that \(\mathcal C\) is either the class of compact convex supports or the (broader) class of compact \(\lambda \)-convex supports (also called \(r\)-convex or \(\alpha \)-convex in the literature). The basic idea is to apply the DB test in a sort of plug-in version, where the support S is approximated by using methods of set estimation. The DBU method is analysed from both the theoretical and practical point of view, via some asymptotic results and a simulation study, respectively.