Tomography

Suppose the function $f$ we must minimize is defined by $$f(x)=h(\sum_{i=1}^n w_ix_i),$$ where $h$ is a convex function of a single variable, and $w$ is a vector of positive numbers.

If $y$ is another vector of $n$ positive numbers we can write $$f(x)=h\left(\sum_{i=1}^n \left(\frac{w_iy_i}{w'y}\right)\left(\frac{w'y}{y_i}x_i\right)\right),$$ and if $g$ is defined as $$g(x,y)=\sum_{i=1}^n\left(\frac{w_iy_i}{w'y}\right)h\left(\frac{w'y}{y_i}x_i\right)$$ then, by the defintion of convexity, $f(x)\leq g(x,y)$. Also, clearly, $f(x)=g(x,x)$ and thus we have a majorization on $(\mathbb{R}^+)^n$.

Alternatively, for any positive vector $\pi$ with elements adding up to one, $$f(x)=h\left(\sum_{i=1}^n\pi_i\left(\frac{w_i}{\pi_i}(x_i-y_i)-w'y\right)\right),$$ and the majorization is $g$ defined by $$g(x,y)=\sum_{i=1}^n\pi_ih\left(\frac{w_i}{\pi_i}(x_i-y_i)-w'y\right).$$