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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.

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    $\begingroup$ What exactly do you mean by that? (convex functions do not form a linear space, etc.) $\endgroup$
    – fedja
    Commented Oct 23, 2017 at 8:41
  • $\begingroup$ @fedja the nonexistence of such bases would answer my question in the negative sense; do you have any references or counterexamples that prove the nonexistence of what I am looking for? $\endgroup$
    – Manfred Weis
    Commented Oct 23, 2017 at 8:50
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    $\begingroup$ Don't you need some extra structure, e.g., $f\in C(\mathbb{R} ^2)$ or $f\in L_2$? For sure you'll need a Hilbert space structure $\endgroup$
    – Amir Sagiv
    Commented Oct 23, 2017 at 9:23
  • $\begingroup$ @AmirSagiv the required structure could be part of an answer; I am looking for bases (resp. decompositions) of continuous convex functions, that would allow for the successive approximation of a convex function that would be the optimal solution of an optimization problem, resp. checking the existence of such a function. $\endgroup$
    – Manfred Weis
    Commented Oct 23, 2017 at 10:05
  • $\begingroup$ So you're looking for an orthonormal basis, such that the projection of any convex function onto the space spanned by that basis is guaranteed to also be convex? $\endgroup$
    – Wouter
    Commented Oct 23, 2017 at 11:05

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