  To a region $C$ of the plane satisfying a suitable convexity condition we
associate a knot concordance invariant $\Upsilon^C$. For appropriate choices of
the domain this construction gives back some known knot Floer concordance
invariants like Rasmussen's $h_i$ invariants, and the Ozsv\'
ath-Stipsicz-Szab\' o upsilon invariant. Furthermore, to three such regions
$C$, $C^+$ and $C^- $ we associate invariants $\Upsilon_{C^\pm, C}$
generalising Kim-Livingston secondary invariant. We show how to compute these
invariants for some interesting classes of knots (including alternating and
torus knots), and we use them to obstruct concordances to Floer thin knots and
algebraic knots.
