Suppose we have a tetrahedron defined by points $(0,0,0),(1,1,0),(0,1,1),(1,0,1)$. Now define surface by $(a,b,a + b - 2ab)$ for $a,b$ between $0$ and $1$. Let $E_1$ be the set of points inside the tetrahedron and $E_2$ be the set of points on that surface, $A$ the area of that surface and $d$ be Euclidian distance
Is it true that for all surfaces $E_3$ with area not greater than $A$?
$$\max_{p \in E1} (\min_{q \in E2} (d(p,q))) \le \max_{p \in E1} (\min_{q \in E3} (d(p,q)))$$
g1 = Graphics3D[{Blue, Opacity[.1], GraphicsComplex[{{0, 0, 0}, {1, 1, 0}, {0, 1, 1}, {1, 0, 1}}, Polygon[{{2, 3, 4}, {3, 2, 1}, {4, 1, 2}, {1, 4, 3}}]]}]; g2 = Plot3D[a + b - 2*a*b, {a, 0, 1}, {b, 0, 1}, PlotStyle -> Opacity[.5]]; Show[g1, g2]