Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S \subseteq P$ be a set of points in $D$.

If $B \in L$ is a block of $D$ and $B \cap S=\{x\}$ then we say that $B$ is tangent to $S$ at $x$.

I am looking for information about sets with following property:

(Ore Property) For every point $ x \in S$ there exists a tangent to $S$ at $x$.

Have such sets been studied?

P.S. Just to be sure, I made up the name of the property, for certain good reasons.

Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S \subseteq P$ be a set of points in $D$.

If $B \in L$ is a block of $D$ and $B \cap S=\{x\}$ then we say that $B$ is tangent to $S$ at $x$.

I am looking for information about sets with following property:

(Ore Property) For every point $ x \in S$ there exists a tangent to $S$ at $x$.

Have such sets been studied?

WHAT I KNOW SO FAR:

I think that this is a weakening of the oval property since I do not require the points to lie on an arc. Apparently there are semiovals which dispense with the arc condition but also require a unique tangent, which is still too strict. So are these

P.S. Just to be sure, I made up the name of the property, for certain good reasons.

Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S \subseteq P$ be a set of points in $D$.

If $B \in L$ is a block of $D$ and $B \cap S=\{x\}$ then we say that $B$ is tangent to $S$ at $x$.

I am looking for information about sets with following property:

(Ore Property) For every point $ x \in S$ there exists a tangent to $S$ at $x$.

Have such sets been studied?

P.S> Just to be sure, I made up the name of the property, for certain good reasons.

Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S \subseteq P$ be a set of points in $D$.

If $B \in L$ is a block of $D$ and $B \cap S=\{x\}$ then we say that $B$ is tangent to $S$ at $x$.

I am looking for information about sets with following property:

(Ore Property) For every point $ x \in S$ there exists a tangent to $S$ at $x$.

Have such sets been studied?

P.S. Just to be sure, I made up the name of the property, for certain good reasons.