Distance to intersection of hyperplane and convex set given another point in convex set and its distance to hyperplane

Question

Let $C\subset \mathbb{R}^m$ be a nonempty, compact and convex set. Moreover, let $\|\cdot\|$ be some norm and $\|\cdot\|'$ its dual norm. That is, for $y\in\mathbb{R}^m$ we define: $$\|y\|'=\max_{x:\|x\|\leq1}\langle x,y\rangle.$$ We now define the hyperplane $Y$ for some $y\in\mathbb{R}^m $ such that $\|y\|'=1$ and $\alpha\in\mathbb{R}$ as: $$Y=\{x\in\mathbb{R}^m:\langle y,x\rangle\leq \alpha\}.$$ Given a $z\in C$ and $\epsilon>0$ such that $\langle y,z\rangle\leq \alpha+\epsilon$, is there always a $x\in C\cap Y$ such that $\|x-z\|\leq\epsilon$?

Answer 1

No. Your assumptions do not guarantee even that $Y \cap C$ is nonempty.

For example in $\mathbb R^1$ and the standard euclidean norm with $y = 1$, $\alpha = 0$ and $C = \{\epsilon\}$. Then $Y = (-\infty, 0]$ (by the way, this is a halfspace, not a hyperplane). And yet $z = \epsilon$ satisfies the condition that $\langle y, z\rangle \leq \epsilon$.