Please consider reopening Planar Quartic Curve invariants for number of connex components. This question was originally unclear because it did not explain that it was working over $\mathbb{R}$, but now it has been edited to be clear.

Let $f_1,\dots,f_m$ be convex functions. Define function $g$ as $$g(x) = \inf_{x_1,\dots,x_m} \{ f_1 (x_1) + \dots + f_m (x_m) \mid x_1+ \dots+x_m = x\}$$ Find $g^*(y)$, the conjugate of $g(x)$.

$A\in R^{m\times n},\ rankA = m,\ f(y) -\text{ Convex function,} \ f:R^m\rightarrow R.$ Let $g(x) = f(Ax)$ Find $g^*(y)$. Because$x\rightarrow Ax$ - Affine transformation $\implies g(x)$ - is convex as well. Because $rank{A}=m \implies g:R^m \rightarrow R$. Because $g(x)$ is convex upwards $-g(...

Problem: Given a direct product $G =C_{r_1} \times C_{r_2} \times \dots \times C_{r_k}$ such that $\sum_{i=1}^k r_i = n$ is the problem input size, what is the minimal expression of $G$ in direct product-of-cyclic groups form, where minimization is with respect to the size function $f(C_{n_1} \ti...