$$\label{e51}
\sup\{\Phi(\Omega,V): ( V:\Omega\rightarrow {\mathbb{R}}^+, \textup{measurable})\}=1,$$ and $$\label{e52}
\inf\{\Phi(\Omega,V): ( V:\Omega\rightarrow {\mathbb{R}}^+, \textup{measurable})\}=\Phi(\Omega).$$ In fact, $$\label{e50}
1>\Phi(\Omega,c) \ge 1-\frac{2^{(m+4)/2}}{|\Omega|}\int_{\Omega}dx\,e^{-c^{1/2}d_{\Omega}(x)/2}, \qquad \forall c\ge 0,$$ where $d_\Omega : \Omega \to {\mathbb{R}}^+$ is the distance function to the boundary, $$d_{\Omega}(x)=\min\{|x-y|:y\in {\mathbb{R}}^m\setminus\Omega\}.$$

We denote the spectrum of $L$ with Dirichlet boundary conditions by $$\{\lambda_1(\Omega,V)\le \lambda_2(\Omega,V)\le \cdots\},$$ accumulating at infinity only. We denote a corresponding $L^2$-orthonormal basis of eigenfunctions by $\{\varphi_{1,\Omega,V}, \varphi_{2,\Omega,V},\cdots\}.$ If the first Dirichlet eigenvalue $\lambda_1(\Omega,V)$ of $L$ has multiplicity $1$, then its corresponding eigenspace is one-dimensional, and $\varphi_{1,\Omega,V}$ is uniquely defined up to a sign. Since $\varphi_{1,\Omega,V}$ does not change sign we may choose $\varphi_{1,\Omega,V}>0.$ If $\lambda_1(\Omega,V)$ is simple, then we define the efficiency of $\varphi_{1, \Omega, V}$ by $$E(\Omega,V)=\frac{\|\varphi_{1, \Omega, V}\|_1}{| \Omega | \, \| \varphi_{1, \Omega, V} \|_{\infty}} .$$ If $V=0$, then we write $$E(\Omega)=E(\Omega,0) \ , \qquad \lambda_1(\Omega) = \lambda_1(\Omega,0).$$ We note that if $\Omega$ is connected, then $\lambda_1(\Omega)$ is simple.

By [@vdBFNT Theorem 1.2] and [@vdB1 Theorem 1] it is possible for $m\ge 2$ to construct, for any $\varepsilon\in(0,1)$, an open connected set $\Omega_{\varepsilon}\subset {\mathbb{R}}^m$ such that both $$\frac{\lambda_1(\Omega_{\varepsilon})\|v_{\Omega_{\varepsilon}}\|_1}{|\Omega_{\varepsilon}|}>1-\varepsilon,$$ and $$\lambda_1(\Omega_{\varepsilon})\|v_{\Omega_{\varepsilon}}\|_{\infty}<1+\varepsilon.$$ This implies that $$\Phi(\Omega_{\varepsilon}) \ge \frac{1-\varepsilon}{1+\varepsilon},\qquad\forall\varepsilon\in(0,1),$$ which in turn implies . We were unable to prove that 