$E(\Omega_{\varepsilon})>1-\varepsilon,\,\forall\varepsilon\in(0,1)$. Nevertheless have the following:

\[the7\] If $m\ge 2$, then $$\sup\{E(\Omega):\Omega\subset {\mathbb{R}}^m,\,\Omega\,\,\,\textup{open and connected},\,0 < |\Omega|<\infty\}=1.$$

By examining the proof of Theorem \[the7\] in Section \[sec5\] we see that for any $m\ge 2$, and $\varepsilon \in (0,1)$ there exists an open, bounded and connected set $\Omega_{\varepsilon}\subset {\mathbb{R}}^m$ such that (i) $E(\Omega_{\varepsilon})\ge 1-\varepsilon$, and (ii) $\lambda_1(\Omega_{\varepsilon})|\Omega_{\varepsilon}|^{2/m} $ is large for $\varepsilon$ small. In Theorem \[the9\] below we show that is a general phenomenon. That is if $\Omega$ is any open and connected set in $ {\mathbb{R}}^m, \,m\ge 2,$ such the mean to max ratio of the first Dirichlet eigenfunction is close to $1$, then the eigenfunction is close to its maximum on most of $\Omega$, and $\lambda_1(\Omega)|\Omega|^{2/m}$ is large. We have a similar phenomenon for the torsion function. Throughout we denote for $\Omega$ open with $0<|\Omega|<\infty$, and $u\in L^1(\Omega),$ $$\fint_{\Omega}u := \frac{1}{|\Omega|} \int_{\Omega} u.$$

\[the9\] Let $m\ge 2$ and let $\Omega$ be a non-empty open set in ${\mathbb{R}}^m$ with finite Lebesgue measure, $|\Omega|<+\infty$.

-   If $u\in H_0^1(\Omega)\cap L^{\infty}(\Omega)\cap\{u\ne 0\}$, and if $$\fint_{\Omega}u\ge \frac{2\|u\|_{\infty}}{m+2}\,$$ then $$\label{p22}
    \bigg(\frac{\omega_m}{|\Omega |}\bigg)^{(2-m)/2}
    \bigg(\|u\|_{\infty}-\fint_{\Omega}u\bigg)\int_{\Omega}|\nabla u|^2\ge \frac{4m^2}{(m+2)^2}\omega_m\|u\|^3_{\infty}.$$ Equality occurs if and only if $\Omega$ is a ball, and $u$ is a multiple of the torsion function.

-   $$\label{p22a}
    