# infimal convolution on conjugate of the sum of convex functions

I came across the following equivalence. Let $$g_j$$ for $$j=1,\ldots,k$$ be closed convex functions. For $$\lambda_j > 0$$ for $$j=1,\ldots,k$$. Then, the term $$\big( \sum_{j=1}^k \lambda_j g_j \big)^*(z)$$ is equivalent to

$$\inf_{\sum z_j = z} \left\{ \sum_{j=1}^k \lambda_j g_j^* \left( \frac{z_j}{\lambda_j} \right) \right\}.$$

It is noted that this is due to the infimal convolution definition applied on the conjugate of the sum of convex functions. I don't see how though. The classical definition as in this Math.SE question is not applicable to this concept, or at least I was not able to show.

How can I show this equivalence, and is there any source online (specifically for this)?

Ok, I solved. here they show that $$(\sum\lambda_j g_j)^* =(\lambda_1g_1)^* \Delta (\lambda_2g_2)^*\Delta \ldots$$ where $$\Delta$$ is the infimal convolution operator.
Next, apply the definition of inf conv to have $$\inf \{ \sum (\lambda_j g_j)^* (z_j) \}$$ or equivalently $$\inf \{ \sum \lambda_j (g_j)^* (z_j/\lambda_j) \}$$ where $$\sum z_j= z$$.