I was reading the paper
Quelques remarques sur les problemes elliptiques quasilineaires du second ordre, P. L. Lions, Journal d’Analyse Mathématique volume 45, pages 234–254(1985)
and on page 251 he claims that (using it in the proof), the distance function near the boundary is concave for a general open bounded domain, which I don't think it is true. I can get this fact if the domain is convex, is there any result saying this for a non-convex domain? Say, interior ball condition is enough?
Edit: On page 251, here is the part I suspected that he used that fact: \begin{align} |u(\overline{x}-s\mathbf{n}(\overline{x}))-u(y)| &= |u(\overline{x}-s\mathbf{n}(\overline{x}))-u(\overline{y} - s\mathbf{n}(\overline{y}))|\\ &\leq \frac{C}{s^{1-\theta}}|\overline{x} - s\mathbf{n}(\overline{x}) - y|. \end{align} My reasoning is that, he has $|\nabla u(x)\leq \frac{C}{d(x)^{1-\theta}}$ earlier. Therefore to connect the line between $\overline{x}-s\mathbf{n}(\overline{x})$ and $y = \overline{y}-s\mathbf{n}(\overline{y})$ we need that every point on this line segment need to have distance bigger or equal to the distance from $\overline{x}-s\mathbf{n}(\overline{x})$ or $y$, in short, we want \begin{equation} d_{\partial \Omega}\Big(\lambda\big(\overline{x}-s\mathbf{n}(\overline{x})\big)+(1-\lambda )y\Big) \geq s \end{equation} provided that at $\lambda = 0$ or $\lambda = 1$ we have \begin{equation} d_{\partial\Omega}(y) = d_{\partial\Omega}(\overline{x}-s\mathbf{n}(\overline{x})) = s. \end{equation} We could say quasi-concave is enough, but I do believe he meant concave.