Let z =k+wt be a simply controlled diffusion process an Rn, where Wtis an n-dimensionnl Brownian motion on (ω,F, P). We say xεRnis stochastically attainable if there exists a eeU, a fixed restraint set, such that P[ZkεBlpar;x)]≥G(k). where B(x) is a closed n-dimoiisional ball of fixed radius centred at x, and G : U-Rnis a given function whose inverse is logarithmically concave. The main result of this paper proves that the stochastic attainable set is convex. This property is useful in establishing the existence and uniqueness of a time-optimal stochastic control.