Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product \(\mathbf f(x)\cdot \mathbf g(y)\) instead of the product of lengths \(|\mathbf f(x)||\mathbf g(y)|\) that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of \(L^p\) norms, also with \(p<1\) , that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato–Ponce inequality (or fractional Leibniz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method.