let $$\label{e11}
\mathfrak{A}((\Omega_n))=\bigg\{(A_n): (\forall n\in {\mathbb{N}})(A_n\subset\Omega_n, A_n\, \textup{measurable}), \lim_{n\rightarrow\infty}\frac{|A_n|}{|\Omega_n|}=0\bigg\}$$ and let $$\label{e12}
\kappa=\sup\bigg\{\limsup_{n\rightarrow\infty}\frac{\int_{A_{n}}v_{\Omega_n,V_n}}{\|v_{\Omega_n,V_n}\|_1}:(A_n)\in\mathfrak{A}((\Omega_n))\bigg\}.$$ We say that $(v_{\Omega_n,V_n})$ $\kappa$-[*localises*]{} if $\kappa>0$, $(v_{\Omega_n,V_n})$ [*localises*]{} if holds with $\kappa=1$.

In Example \[exa2\] in Section \[sec4\] we analyse a sequence of open sets in ${\mathbb{R}}^2$ where localisation occurs for certain parameter choices. For one specific parameter choice the sequence of torsion functions is $\kappa$-localising with $0<\kappa<1$.

The theorem below asserts that localisation implies vanishing efficiency.

\[the5\] Let $(\Omega_n)$ be a sequence of open sets in ${\mathbb{R}}^m$ with $0 < |\Omega_n|<\infty$ and let $V_n:\Omega_n\to {\mathbb{R}}^+$ be a sequence of measurable functions. If $(v_{\Omega_n,V_n})$ localises, then $(v_{\Omega_n,V_n})$ has vanishing efficiency, $\lim_{n\rightarrow\infty}\Phi(\Omega_n,V_n)=0.$

It follows from Theorem 1.1 in [@DPGGLB] that, when restricting $\Omega$ to be open, bounded and convex in ${\mathbb{R}}^m,\, m\ge 1$, one has $$\label{e70}
\frac{2}{m(m+2)}\le \inf\{\Phi(\Omega): \Omega\subset {\mathbb{R}}^m,\,\Omega\,\,\,\textup{open,\,convex},\,|\Omega|<\infty\}.$$ We see by that any sequence $(\Omega_n)$ of elongating open, bounded and convex sets in ${\mathbb{R}}^m$ has non-vanishing efficiency, and so by Theorem \[the5\], $(v_{\Omega_n,0})$ is not localising. This is in contrast with the results of [@vdBBDPG], where localisation of the first Dirichlet eigenfunction in $L^2(\Omega_n)$ was obtained for a wide class of elongating, open, bounded, convex sets in ${\mathbb{R}}^m$. Further examples demonstrating different behaviour of the 