Inflection points and double tangents on anti-convex curves in the real projective plane

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delta(\gamma)=3$ for anti-convex curves, where $i(\gamma)$ is the number of independent (true) inflection points and $\delta(\gamma)$ the number of independent double tangents. This formula is a refinement of the classical M\"obius theorem. We shall also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.