I thought this result was a bit interesting. Mahlon M. Day in the paper [1] showed that the amenable groups are precisely the groups where there Markov-Kakutani theorem holds.
If $(X,\mathcal{M})$ is an algebra of sets, then a function $\mu:\mathcal{M}\rightarrow[0,1]$ is said to be a finitely additive probability measure if $\mu(\emptyset)=0,\mu(X)=1$ and $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A,B\in\mathcal{M}$ and $A\cap B=\emptyset$. If $G$ is a group, then a finitely additive probability measure $\mu:P(G)\rightarrow G$ on the algebra of sets $(G,P(G))$ is said to be left-invariant if $\mu(aR)=\mu(R)$ for each $R\subseteq G$.
A group $G$ is said to be amenable if there exists a left-invariant finitely additive probability measure $\mu:P(G)\rightarrow[0,1]$. For example, every finite group is amenable, and every abelian group is amenable. Furthermore, the class of amenable groups is closed under taking quotients, subgroups, direct limits, and finite products.
Let $C$ be a convex subset of a real vector space. Then a function $f:C\rightarrow C$ is said to be an affine map if $f(\lambda x+(1-\lambda)y)=\lambda f(x)+(1-\lambda)f(y)$ for each $\lambda\in[0,1]$ and $x,y\in C$.
$\textbf{Theorem}$(Day) Let $G$ be a group. Then the following are equivalent.
$G$ is amenable.
Let $X$ be a Hausdorff topological vector space and let $C\subseteq X$ be a compact convex subset. Let $\phi:G\rightarrow C^{C}$ be a group action such that each $\phi(g)$ is a continuous affine map. Then there is a point in $C$ fixed by every element of $G$.
Let $X$ be a locally convex topological vector space and let $C\subseteq X$ be a compact convex subset. Let $\phi:G\rightarrow C^{C}$ be a group action such that each $\phi(g)$ is a continuous affine map. Then there is a point in $C$ fixed by every element in $G$.
[1] Fixed-point theorems for compact convex sets. Mahlon M. Day.Illinois J. Math. Volume 5, Issue 4 (1961), 585-590.
[2] Ceccherini-Silberstein, Tullio, and M. Coornaert. Cellular Automata and Groups. Heidelberg: Springer, 2010.