I'm trying to find an interpolation for a continuous real function $f:\Bbb [0,1]^3\rightarrow \Bbb R$ from the values it takes at the eight points $\{0,1\}^3$ (the vertices of a unit cube); I have the following restraints:
- if a point $p$ is on a face of the cube ($p\in [0,1]^3-(0,1)^3$), the interpolated value at that point should only be determined by the four vertices of that face;
- $|\nabla f|=1$ where $\nabla f$ is defined;
- $\nabla f$ is defined almost everywhere
- $\forall\, u,v \in [0,1]^3\ \ |f(u)-f(v)|\le|u-v|$ (this is very probably deducible from point 2, but I'm not 100% sure so I put it anyway); something that might be useful to know is that this makes it possible to find higher and lower bounds to the value of the function (copying the derivation here would make this a bit longer, so I won't, but if you think I should please tell me in a comment)
- $\forall u \in [0,1]^3 \; f(u-f(u)\nabla f(u))=0$, if $f(u-f(u)\nabla f(u))$ is defined (this is not required but if you find a way to enforce it it will make my work down the road much easier);