Let $H$ be a Hilbert space, $e\in H$ with $\|e\|=1$. Let $L=\{x\in H\mid \langle x,e\rangle=0\}$. For $L$ we consider the sublinear function $d_L\colon H\to \mathbb{R}$ $$d_L(h):=\inf\{\|h-y\| : y\in L\}.$$ Claim: $d_L(h)=|\langle h,e\rangle|$ for all $h\in H$.
I first thought about applying Riesz representation theorem for Hilbert spaces but here we have sublinear functions instead of linear functionals (i.e. instead of elements in the dual $H'$) and I'm not sure how to define a suitable functional to apply this theorem.
An attempt: Is is $min\{\|x-y\|:y\in L\}=max\{|\langle x,y\rangle|:y\in L^\perp , \|y\|=1\}$ (see for example Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$ ) We have that $e\in L^\perp$. If it is possible to show that $max\{|\langle x,y\rangle|:y\in L^\perp , \|y\|=1\}=|\langle x,e\rangle|$, we are done. But I don't know how to prove the last equlity.
How to prove $$max\{|\langle x,y\rangle|:y\in L^\perp , \|y\|=1\}=|\langle x,e\rangle|$$ or how else to prove the claim? I appreciate any help and hints.