A function $f:D\to \mathbb{R}$ is said to be symmetrically $\gamma$-convex w.r.t. the roughness degree $\gamma >0$ if the Jensen inequality $$f(x_{\lambda })\le (1-\lambda )f(x_0)+\lambda f(x_1),x_{\lambda }:=(1-\lambda )x_0+\lambda x_1$$ is fulfilled for all $x_0,x_1\in D$ satisfying $\Vert x_0-x_1\Vert \ge \gamma$ and for $$\lambda ={\displaystyle \frac{\gamma }{\Vert x_1-x_0\Vert }}\text{and}\lambda =1-{\displaystyle \frac{\gamma }{\Vert x_1-x_0\Vert }}.$$ Such a function also has some analytical properties which are similar to those of convex functions. For instance, if it is bounded above on some sphere $\{x\in X:\Vert x-x^{\ast }\Vert =\gamma \}\subset D$ then it is bounded on the ball ${\bar{\mathcal{U}}}_{\gamma }(x^{\ast }):=\{x\in X:\Vert x-x^{\ast }\Vert \le \gamma \}$ and bounded below on each bounded subset of $D$. If the domain $D$ is so large that its interior contains some ball ${\bar{\mathcal{U}}}_{\gamma }(x^{\ast })$, and if the symmetrically $\gamma$-convex function considered is locally bounded above at some interior point of $D$, then it is locally bounded in the interior of $D$.