Fig. 1:

Verification of proposition 1 in the 2D and 3D. The 2D curve $\mathcal{C}$ is the zero level set of ψ, whose bandwidth is 3×3. By Proposition 1, we need at least 8 points to recover the curve. Curve recovery failed from a random choice of 7 points (red dots) in (b), while another random set of 8 points succeeded in (c), yielding perfect recovery. For the 3D case, we choose the surface to be the zero level set of a polynomial of bandwidth 3×3×3. Our conclusion requests at least 26 points to recover the surface. (e) and (f) illustrated the conclusion.