  We propose a procedure for the $D\rightarrow 4$ limit of
Einstein-Gauss-Bonnet (EGB) gravity that leads to a well defined action
principle in four dimensions. Our construction is based on compactifying
$D$-dimensional EGB gravity on a $(D-4)$-dimensional maximally symmetric space
followed by redefining the Gauss-Bonnet coupling $\alpha\rightarrow
\frac{\alpha}{D-4}$. The resulting model is a special scalar-tensor theory that
belongs to the family of Horndeski gravity. Static black hole solutions in the
scalar-tensor theory are investigated. Interestingly, the metric profile is
independent of the curvature of the internal space and coincides with the
$D\rightarrow 4$ limit of the usual EGB black hole with the unusual
Gauss-Bonnet coupling $\frac{\alpha}{D-4}$. The curvature information of the
internal space is instead encoded in the profile of the extra scalar field. Our
procedure can also be generalized to define further limits of the Gauss-Bonnet
combination by compactifying the $D$-dimensional theory on a
$(D-p)$-dimensional maximally symmetric space with $p\leq 3$. These lead to
different $D\rightarrow 4$ limits of EGB gravity as well as its $D\rightarrow
2,3$ limits.
