What's this "Euler's formula"? 
convex. Indeed, differentiating Euler's foruma $\langle F_x,x\rangle=F$ gives $F_{xx}x=0$. By  the convexity assumption $F_{xx}\,\Big|\,TS>0$. Therefore, we define the function

From Hofer-Zehnder, p. 25, preliminaries to proof of thm 5 in paragraph 1.5. $F$ is a function in $\mathbb{R}^{2n}$, defined in the following way. $S$ is a strictly convex hypersurface. We can assume the interior of the region $S$ is the boundary of contains the origin. Then given $x\neq0$ the ray from the origin to $x$ intersects $S$ in a single point $\xi$, which we define to be $\lambda^{-1}x$. Then we set $F(x)=\lambda$, so that $S=\{x:F(x)=1\}$. That's the context. What is he referring to as «Euler's formula»? What is $\langle F_x,x\rangle$? How do I prove the formula?

I am not sure if this is what is meant here but there is Euler's theorem on homogenous functions which says that given function homogenous of rank rr (I think there is restriction on r but don't remember exactly) that is $f(\lambda x) = \lambda ^r f(x)$, we have $r f(x)=x \cdot \nabla f (x)$. This looks like your formula with $r=1$ and $F_x$ denoting partial derivative w.r.t. x (or gradient if x is a vector) and bracket denoting scalar product. Does this interpretation make sense in this context?
