Confusion about the convexity of the best response correspondence
I would have thought that the if the opponent strategy $\sigma_{-i}$ is given, even if it is a mixed strategy, then $$u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})= \lambda u_i(\sigma_i, \sigma_{-i})+(1-\lambda)u_i(\sigma_i', \sigma_{-i})$$ and so if  $u_i(\sigma_i, \sigma_{-i})=u_i(\sigma_i', \sigma_{-i})$ then $u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})$ is equal to both of them. If they are best responses then so too is the linear combination strategy