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Graph convexity spaces have been studied in many contexts. In particular,some studies are devoted to determine if a graph equipped with a convexityspace is a {\em convex geometry}. It is well known that chordal and Ptolemaicgraphs can be characterized as convex geometries with respect to the geodesicand monophonic convexities, respectively. Weak polarizable graphs, intervalgraphs, and proper interval graphs can also be characterized in this way. Inthis paper we introduce the notion of {\em $l^k$-convexity}, a naturalrestriction of the monophonic convexity. Let $G$ be a graph and $k\geq 2$ aninteger. A subset $S\subseteq V(G)$ is \textit{$l^k$-convex} if and only if forany pair of vertices $x,y$ of $S$, each induced path of length {\em at most}$k$ connecting $x$ and $y$ is completely contained in the subgraph induced by$S$. The {\em $l^k$-convexity} consists of all $l^k$-convex subsets of $G$. Inthis work, we characterize {\em $l^k$-convex geometries} (graphs that areconvex geometries with respect to the $l^k$-convexity) for $k\in\{2,3\}$. Weshow that a graph $G$ is an $l^2$-convex geometry if and only if $G$ is achordal $P_4$-free graph, and an $l^3$-convex geometry if and only if $G$ is achordal graph with diameter at most three such that its induced gems satisfy aspecial "solving" property. As far as the authors know, the class of$l^3$-convex geometries is the first example of a non-hereditary class ofconvex geometries.

Convex geometries over induced paths with bounded length