Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the polytope covering the convex hull of the points.
Let $C$ be any compact, closed space containing a subset of $P$.
Let $$S=\{f\in K[x_1,\dots,x_n]/I:\mathsf{Z}(f)\subset C\} (\mbox{ zero set contained inside }C)$$
$$d_{}=\min\{deg(f):f\in S_{}\}$$ $$\bar{d}_{}=\min\{deg(f):f\notin S_{}\}$$ $${D}_{}=\min\{deg(f):(f\in S)\mbox{ and }(f(z)=1\mbox{ }\forall z\in Z(I)\backslash C)\}$$ $$\bar{D}_{}=\min\{deg(f):(f\notin S)\mbox{ and }(f(z)=1\mbox{ }\forall z\in Z(I)\cap C)\}$$
Is $d\bar{d}\leq D\bar{D}\leq(d\bar{d})^2\leq (D\bar{D})^2$?
The first and the last inequalities are clear. Is the second inequality also true?