Minima of quasisuperminimizers

Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. Then u is a quasisuperminimizer, and we improve upon the known upper bound (due to Kinnunen and Martio) for the optimal quasisuperminimizing constant Q of u. We give the first examples with Q>Q_2, and show that in general Q>Q_2 whenever Q_1 >1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.