Let D be a bounded Lipschitz domain in \({\mathbb{R}}^ n\), \(n\geq 3\). The author proves the following extensions of results of \textit{J. Lelong- Ferrand} [Ann. Sci. Éc. Norm. Supér., III. Sér. 66, 125-159 (1949; Zbl 0033.373)]. If \(E\subseteq D\) and E is thin at \(\xi\in \partial D\), then E is minimally thin at \(\xi\). Conversely, if E is contained in a nontangential cone with vertex at \(\xi\in \partial D\), and E is minimally thin at \(\xi\), then E is thin at \(\xi\). He also considers Green potentials with \(L^ p\)-densities. Let \(G(x,f)=\int_{D}G(x,y) f(y) dy\) and \(G(x,\mu)=\int_{D}G(x,y) d\mu (y)\) for a nonnegative function f and measure \(\mu\) on D, and let \(p\geq 1\). If G(\(\cdot,f)\) and G(\(\cdot,\mu)\) are non-trivial, and \[ \int_{D}f(x)^ p \delta (x)^{2p-1} dx1,\quad \int_{D}\delta (x) d\mu (x)<\infty \quad if\quad p=1, \] where \(\delta\) denotes the distance from \(\partial D\), it is shown that G(\(\cdot,f)\) and G(\(\cdot,\mu)\) have certain boundary limits outside a set of (n-1)dimensional Hausdorff measure zero.