$\delta$-convexity in normed linear spaces

For some given positive $\delta$, a function $f:D\subseteq X\to\RR$ is called $\delta$-convex if it satisfies the Jensen inequality $f(x_\lambda)\leq(1-\lambda)f(x_0)+\lambda f(x_1)$ for all $x_0, x_1\in D$ and $x_\lambda\colon=(1-\lambda)x_0+\lambda x_1 \in [x_0,x_1]$ satisfying $\|x_0-x_1\|\ge\delta$, $\|x_\lambdax _0\|\ge\delta/2$ and $\|x_\lambda-x_1\|\ge\delta/2$. In this paper, we introduce $\delta$-convex sets and show that a function $f:D\subseteq X\to\RR$ is $\delta$-convex iff the level set $\{x\in D:f(x)+\xi(x)\leq \alpha\}$ is $\delta$-convex for every continuous linear functional $\xi\in X^*$ and for every real $\alpha$. Some optimization properties such as constant property on affine sets, and analytical properties such as boundedness on bounded sets, local boundedness, conservation and infection of $\delta$-convex functions are presented.