Abstract: The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bézier nets and Bernstein polynomials over a simplex σ is an angle φ determined by σ, and proves that (1) if f(x) is Lipschitz continuous over σ, i.e., f(x) ∈ Lip_A(α,σ), then both the n-th Bézier net f_n and the n-th Bernstein polynomial B_n(f; x) corresponding to f(x) belong to Lip_B(α,σ), where B=Asec^αφ; and (2) if n-th Bézier net f_n ∈ Lip_A(α,σ), then the elevation Bézier net Ef_n and the corresponding Bernstein polynomial B_n(f,;x) also belong to Lip_A(α,σ). Furthermore, the constant B=Asec^αφ in case (1) is best in some sense.

Key words: Bernstein polynomials, BBézier nets, shape preserving property, Lipschitz continuity, simplex