Given a bounded domain Ω in Rn and p in ]1, ∞[, the Lp Hardy constant Hp is the best constant for the Hardy inequality in W1,p(Ω). In this talk we present monotonicity, continuity and differentiability results concerning the dependence of Hp on p. Time permitting, we shall also discuss the dependence of such constant on Ω. The focus is on non-convex domains in which case the value of Hp is in general not explicitly known.