We prove the modular convexity of the mixed norm L p (ℓ 2) on the Sobolev space W 1 , p (Ω) in a domain Ω ⊂ R n under the sole assumption that the exponent p (x) is bounded away from 1, i.e., we include the case sup x ∈ Ω p (x) = ∞ . In particular, the mixed Sobolev norm is uniformly convex if 1 < inf x ∈ Ω p (x) ≤ sup x ∈ Ω p (x) < ∞ and W 0 1 , p (Ω) is uniformly convex.