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 $$f(x_{\lambda})\leq (1-\lambda)f(x_0)+\lambda f(x_1), \quad x_{\lambda}:=(1-\lambda)x_0+\lambda x_1$$ is fulfilled for all $x_0, x_1 \in D$ satisfying $\|x_0-x_1\|\geq \gamma$ and for $$\lambda=\dfrac{\gamma}{\|x_1-x_0\|}\quad \text{and}\quad \lambda=1-\dfrac{\gamma}{\|x_1-x_0\|}.$$ 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 $\{x\in X\,:\, \|x-x^*\|=\gamma\}\subset D$ then it is bounded on the ball $\overline{\mathcal{U}}_{\gamma}(x^*):=\{x\in X\,:\, \|x-x^*\|\leq \gamma\}$ and bounded below on each bounded subset of $D$. If the domain $D$ is so large that its interior contains some ball $\overline{\mathcal{U}}_{\gamma}(x^*)$, 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 $D$.