Are there any orthonormal bases for strictly convex functions $f: \mathbb{R}^n\ni x \mapsto \mathbb{R},\ x\ne y\implies f\left(\alpha x+\left(1-\alpha\right)y\right) \lt \alpha f(x)+(1-\alpha)f(y) \wedge \alpha\in(0,1)$?   

The subset of $\mathbb{R}^n$ on which $f$ is defined can be restricted in any appropriate way e.g. unit cube, unit sphere, etc.

edit:  
what I want to do, is to assign elements $p\in\mathbb{R}^n$ to the vertices of a complete, finite metric graph $G(V,E)$ in a way that the $\left(p_i,\sum_{V\setminus i}w(e_{ij})\right)\in\mathbb{R}^{n+1}$ are in convex configuration, and to determine a convex function $f:\mathbb{R}^n\ni x\mapsto \mathbb{R}$ with $f(p_i)=\sum_{V\setminus i}w(e_{ij})\ \forall i,j\quad \wedge\quad \sum\|p_j-p_i\|=1\quad \wedge\quad \min_{i\ne j}{\|p_j-p_i\|}=max\quad$ for some fixed $2\le n\le |V|$; those conditions do not yet rule out ambiguity w.r.t. isometric transformation of the points, but that can be easily fixed.