In this short note two unconventional overdetermined problems are considered. Let p ∈ (1, n); first, the following is proved: if Ω is a bounded domain in R n whose p-capacitary potential function u has two homotetic convex level sets, then Ω is a ball. Then, as an application, we obtain the following: if Ω is a convex domain in R n whose p-capacitary potential function u is (1 − p)/(n − p)-concave (i.e. u (1−p)/(n−p) is convex), then Ω is a ball.