> 
> **Given a twice-differentiable function $f: \mathbb{R}^n \to \mathbb{R}$, $f$ is convex iff $\nabla ^2 f(x) \succeq 0, \forall x \in \text{dom} f$.**
> 
> 
> 

This is the definition given on these sets of notes <https://www.seas.ucla.edu/~vandenbe/ee236b/lectures/functions.pdf>

But I feel like something is missing in the definition...

What does it mean for $\nabla ^2 f(x) \succeq 0$? 

**Does it mean:**

1. $y^T\nabla ^2 f(x)y \geq 0, \forall y \in \text{dom} f$ or...
2. $y^T\nabla ^2 f(x)y \geq 0, \forall y \in \mathbb{R}^n$