I want to find convex conjugate of $f(x) = (\max_{i \leq n}x_i)(-\sum \log x_i)$ where $x \in \mathbb{R}^n_{++}$. This function looks like the negative entropy function but there is $(\max_{i \leq n}x_i)$ instead of $x_i$.
It seems like $f^*(y) = \sup_x \{xy - f(x)\}$ is unbounded above for any $y$. I think so because I first tried to find conjugate of $g(x) = -\sum x_ilogx_i$. Then $x_iy_i+x_ilogx_i$ is unbounded for every $i$ even when all $y_i < 0$. The function in the title is even greater then negative entropy, so I thought that it is unbounded too. So what is conjugate of this function if the domain is empty?
The problem is that your function is not convex. It is not concave either. This can be seen for $n=2$ by changing just one variable: $g(x) = f(x,y) = -\max\{x,y\}\cdot(\log x + \log y)$. The function $g$ is concave when $x>y>0$ and convex when $y>x>0$.