Lipschitz smoothness, strong convexity and the Hessian

Question

I'm working with the following two concepts:

• Lipschitz Smoothness - a function $f$ is Lipschitz smooth with constant $L$ if its derivatives are Lipschitz continuous with constant $L$, in other words if for any $x$ and $y$, $$ \| \nabla f(x) - \nabla f(y) \| \leq L \| x - y \| $$
• Strong Convexity - a function $f$ is $\alpha$-strongly convex if $$ \nabla^2 f(x) \succeq \alpha I $$ for all $x$, where $I$ is the identity matrix.

Here are my questions:

1. I know that Lipschitz Smoothness implies that for any $x$ and $y$, it is true that $$ f(x + y) \leq f(x) + y^\top \nabla f(x) + \frac{L}{2} \| y \|^2 $$ Is the converse also true? ie: is it an "if and only if"?
2. I read somewhere that $f$ is Lipschitz smooth if and only if $$ L I \succeq \nabla^2 f(x) $$ for all $x$, where $I$ is the identity matrix. How can I prove that?
3. I read somewhere that $f$ is $\alpha$-strongly convex if and only if for any $x$ and $y$, it is true that $$ f(x+y) \leq f(x) + y^\top\nabla f(x) - \frac{\alpha}{2} \| x - y \| $$ How can I prove this, and is it and if-and-only-if?
4. The previous two points seem to imply some relationship between these two concepts. Does it go any deeper?

I realize this is a bunch of questions - if anyone has a good reference on these topics, that'd be swell too.

Answer 1

Regarding the references, Nesterov's "Introductory Lectures on Convex Optimization" (ISBN: 978-1-4613-4691-3) answers at least three out of the four questions asked here. Specific equations/theorems:

Question 1: See equation (1.2.12) and substitute "y" for "x+y"

Question 2: Theorem 2.1.6. with adjoining proof.

Question 3: As mentioned in the previous answer, the equation was slightly wrong in the original question. But if corrected, this is proven in Theorem 2.1.11.