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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.

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  • $\begingroup$ I think the present wording is unclear. Is this a property of S, B, or D? (I'm guessing S.) $\endgroup$
    – The Masked Avenger
    Mar 23, 2014 at 19:49
  • $\begingroup$ @TheMaskedAvenger Of $S$. $\endgroup$
    – Felix Goldberg
    Mar 23, 2014 at 21:00
  • $\begingroup$ You probably know this already, but $S$ has your Ore property whenever $S$ is contained in a proper flat, and also whenever $P$ is large relative to (quadratic in?) $S$. $\endgroup$
    – Peter Dukes
    Mar 25, 2014 at 21:50
  • $\begingroup$ @PeterDukes Actually, I don't so happy to learn something new! :) But how do you define a flat in a general design? $\endgroup$
    – Felix Goldberg
    Mar 25, 2014 at 22:57
  • $\begingroup$ Flats are also known as subdesigns, or subspaces. By the way, I am interested in designs such that any "small" collection of points is contained in a proper flat/subdesign. They exist with surprising abundance, not just in affine or projective space of high dimension. There are lots of tangents in these cases. $\endgroup$
    – Peter Dukes
    Mar 26, 2014 at 0:21

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