Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits\_{l\in \{a,b\}}\{d(x,l)\}$.
Let $(X,\*)$ be a topological group with the following property:
$$h(xy)\leq \max\{h(x),h(y) \}.$$
It is a locally compact, connected, simply connected, Hausdorff group. I think it might be a Lie group, but which one I don't know. This situation can be transferred to higher dimensions and there I am also interested in the next question.
Does such a group exist?



Group structure on the strip
Such group does not exist. To derive a contradiction, assume that the strip $X=\mathbb R\times(-1,1)$ admits a continuous group operation $X\times X\to X$, $(x,y)\mapsto xy$, such that $h(xy)\le\max\{h(x),h(y)\}$ for all $x,y\in X$.


Let $c$ be any point on the central line $L=\mathbb R\times\{0\}$ and $f:X\to X$ be the homeomorphism defined by $f(x)=x^{-1}c$. Observe that $xf(x)=xx^{-1}c=c\in L$ for every $x\in X$.


Since the line $L$ is nowhere dense in $X$, the set $L\cup f^{-1}[L]$ is nowhere dense in $X$ and hence we can choose an element $x\in X\setminus (L\cup f^{-1}[L])$. Then for the points $x\notin L$ and $y=f(x)=x^{-1}c\notin L$ we have
$$1=h(c)=h(xy)>\max\{h(x),h(x^{-1}c)\},$$
which contradicts the choice of $h$.


