I am reading the paper 'A Convexity Inequality' by Roselli-Willem. It is a wonderful article. There is one small part that I am not able to figure out. Could you please help me?
Let $J:[0,\infty)\times [0,\infty)\to\mathbb{R}$ be continuous and positively homogeneous (i.e. $J(rx,ry)=r\, J(x,y)$, for all $r,x,y\in [0,\infty)$). Given that the function $\varphi:[0,\infty)\to\mathbb{R}$ defined as$$\varphi(t):=J(1,t),\text{ for all }t\in [0,\infty),$$is convex, I want to show that
CLAIM: there exists a $\Gamma\subset \mathbb{R}^2$ such that$$J(x,y)=\sup\{ay+bx:(a,b)\in\Gamma\}, \text{ for all }(x,y)\in [0,\infty)\times [0,\infty). \ \ \ (***)$$
To prove this, what I tried is follows. Since $\varphi$ is convex, we can find a $\Gamma\subset \mathbb{R}^2$ such that$$\varphi(t)=\sup\{at+b:(a,b)\in\Gamma\}, \text{ for all }t\in [0,\infty).$$We shall show that the same $\Gamma$ works in $(***)$.
When $x>0$, $(***)$ follows from homogeneity of $J$, i.e. $J(x,y)=x\varphi\left(\frac{y}{x}\right)$. However, I am unable to show $(***)$ when $x=0$. In other words, I am unable to prove that$$J(0,1)=\sup\{a:(a,b)\in\Gamma\}.$$Could you please suggest how to prove it? Thank you.