A convex set is
\mathcal{F}
F
-convex if every pair of points in the set lie in a right triangle included in the set. We characterize
\mathcal{F}
F
-convex sets, find some classes of
\mathcal{F}
F
-convex sets, investigate
\mathcal{F}
F
-convexity for cones and cylinders, and find out that most convex bodies are
\mathcal{F}
F
-convex. On our way, we also describe the curvature at the endpoints of diameters of most convex bodies