Distance between a point $x\in H$ and $H\cap \overline{B}(0,1)$, with $H$ a hyperplane Let $X$ be a normed space. Let $f:X\to\mathbb{R}$ be a linear functional, and consider the hyperplane $H=f^{-1}(r)$, for some $r>0$.
Let  $x\in H$ be such that $1<\left\|{x}\right\|\le 1+\epsilon$, for some $\epsilon>0$.
If $B$ is the unit closed ball and $H\cap B\neq\emptyset$, my question is:

Is it true that $d(x,H\cap B)\le\epsilon$ ?

In any case, I am trying to prove that $d(x,H\cap B)$ has to be "small", so $2\epsilon$ instead of $\epsilon$ would do as well. But I'm still unable to find a suitable $y\in H\cap B$ such that $d(x,y)$ is small. In $\mathbb{R}^2$ (visually speaking), it becomes obvious that such a $y$ has to exist, but I'm stuck when $X$ is any normed space.

The answer is no, and here is a counter example in $X=\mathbb R²$, with the norm $\|(x,y)\|=\max\{|x|,|y|\}$.

