torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima for elongated convex planar domains have been constructed in [@Beck].

Theorem \[the8\] below, asserts that (under suitable hypotheses), vanishing efficiency for the torsion function implies localisation for the first Dirichlet eigenfunction. We make two further definitions.

\[def4\] Let $\Omega \subset {\mathbb{R}}^m$ be an open set with $0 <|\Omega|<\infty$, and let $V:\Omega\rightarrow {\mathbb{R}}^+$ be measurable, such that $\lambda_1(\Omega,V)$ has multiplicity $1$. Let $0 < \kappa<1,\, 0< \delta < 1.$ We say that $\varphi_{1, \Omega,V}$ $(\kappa, \delta)$-localises if there exists a measurable subset $A \subset \Omega$ with $$\frac{|A|}{|\Omega|} < \delta,\qquad
\frac{\int_A \varphi_{1, \Omega,V}}
{\|\varphi_{1, \Omega, V}\|_1}
\ge \kappa.$$

Following Definition \[def3\] we also introduce the notion of localisation for a sequence of first Dirichlet eigenfunctions.

\[def5\] Let $(\Omega_n)$ be a sequence of open sets in ${\mathbb{R}}^m$ with $0 < |\Omega_n|<\infty,\,n\in{\mathbb{N}}$, and let $V_n:\Omega_n\rightarrow{\mathbb{R}}^+,\,n\in {\mathbb{N}}$, be measurable, such that $\lambda_1(\Omega_n,V_n)$ has multiplicity $1$ for all $n\in {\mathbb{N}}$. Let $\mathfrak{A}((\Omega_n))$ be given by , and let $$\label{extra}
\kappa=\sup\bigg\{\limsup_{n\rightarrow\infty}\frac{\int_{A_{n}}\varphi_{1, \Omega_n,V_n}}{
\|\varphi_{1,\Omega_n,V_n}\|_1}:(A_n)\in\mathfrak{A}((\Omega_n))\bigg\}.$$ We say that $(\varphi_{1,\Omega_n,V_n})$ $\kappa$-[*localises*]{} if $\kappa>0$, $(\varphi_{1,\Omega_n,V_n})$ [*localises*]{} if holds with $\kappa=1$.

\[the8\] Let $(\Omega_n)$ be a sequence of open subsets in ${\mathbb{R}}^m$ with $0 < |\Omega_n| < \infty$, 