What is the cheapest known finite dimensional approximation of Lipschitz functions.

Question

Let $Lip_{1}([0,1]^{d})$ the set of all Lipschitz functions on $[0,1]^{d}$ with Lipschitz constant less or equal to $1$. I would like to approximate the set with respect to the uniform topology by a finite dimensional subset (like Bernstein polynomials for Lipschitz functions on $[0,1]$). Preferably I would like the approximating set to also be convex. For a specific degree $\epsilon>0$ of approximation, what is the cheapest, in terms of dimension of the approximating space, known way to do it? Also, does the space of all Lipschitz functions have a Shauder basis?