There is a well known result that $\frac{\partial^2 c(K)}{\partial K^2}= e^{-rT}f(K)$, where $K$ is the strike and $f$ is the risk neutral density at time $T$. With the left calculated from market prices, this formula can be used to obtain the implied risk neutral density.
However, what happens if the market prices are such that this doesn't give a valid density. For example, the market may give you a $\frac{\partial^2 c(K)}{\partial K^2}$ such that $f(K)$ doesn't integrate to unity.
Are there any known conditions on the implied volatility or $c(K)$ such that $f(K)$ is a valid density?