Weakly affine functions

Let $\emptyset\neq D\subseteq\mathbb{R}^n$ be a convex set. We say that the function $f:D\longrightarrow\mathbb{R}$ is weakly $k$-affine if, for any affinely independent system of $k+1$ points $x_0, x_1,\dots,x_k\in D$, there exist $\lambda_0,\lambda_1,\dots,\lambda_k\in\left] 0,1\right[$ such that $\lambda_0+\lambda_1+\dots+\lambda_k=1$ and $$f\left(\lambda_0x_0+\lambda_1x_1+\dots+\lambda_kx_k\right)=\lambda_0f(x_0)+\lambda_1f(x_1)+\dots+\lambda_kf(x_k).$$ Our main result is that any continuous, weakly $2$-affine function is necessarily affine. It is a well-known result that a continuous, weakly $1$-affine function is affine. However, in the paper, we will show that if continuity is replaced by several weaker regularity conditions, then this implication fails to hold. We also introduce a new concept of generalized convexity, namely the class of weakly $k$-convex sets, which turns out to be naturally related to weakly $k$-affine functions. We present results concerning a subclass of continuous, weakly $k$-affine functions for $k\geq 3$, as well.