The usual definition of semiconcave for a function $f$ defined on an open set $\Omega$ is, if for any $x\in \Omega,h\in \mathbb{R}^n$ such that $x,x-h,x+h\in \Omega$ then $$ f(x-h)-2f(x) + f(x+h) \leq c|h|^2$$ for some $c>0$.
My question is, would it be enough to have $$ f(x-h)-2f(x) + f(x+h) \leq c|h|^2$$ for all $|h|\ll 1$ small enough and still have $f$ is semiconcave?