Why is the lagrange dual function concave?

The Lagrange function $L(x,\lambda,\mu)$ is affine in $\lambda$ and $\mu$, thus, concave, and the infimum of concave functions is concave. (Equivalently to the supremum of convex is convex.)

The book referenced is Convex Optimization by Boyd and Vandenberghe. To better see the "pointwise infimum", consider a slight change/abuse of notation: $L_x(\xi) = L(x, \lambda, \nu)$ where $\xi = (\lambda, \nu)$. For a fixed $x$, $L_x(\xi)$ is affine in $\xi$ so $\{L_x(\xi) \,:\, x \in \mathcal{D}\}$ is a family of affine functions and its pointwise infimum is $$g(\xi) = \inf_x \,\{L_x(\xi)\,:\,x\in\mathcal{D}\}$$ Now we can use @A.Γ.'s pointer to show that $g$ is concave by showing the epigraph of $-g$ is convex. For a given $\xi$, we have $g(\xi) \le L_x(\xi)$ for any $L_x$ from the family so $(\xi, -g(\xi))$ is always "above" $(\xi, -L_x(\xi)$) hence $\rm{epi}(-g) \subset \bigcap_x \rm{epi}(-L_x)$.