Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

Question

I am reading the paper of Prof. Yurii Nesterov:
Primal-dual subgradient methods for convex problems

I am confused about the green part of the following:

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1. Why it is enough to show the boundedness of level sets to ptove the existence of a solution?

2. Why (9.3) can tell you the solution is unique?

Answer 1

1. $d$ is continuous, thus the level set is closed. If it is also bounded, it is compact.

2. Strong convexity implies strict convexity. Strictly convex function has at most one minimum.

   Assume $y\ne z$ are both minima. Then, by strict convexity we have $$ f\left(\frac{y+z}{2}\right) < \frac12\left(f(y) + f(z)\right).$$ That contradicts that $y$ and $z$ are minima.

Answer 2

Then problem becomes $\min_x f(x)$ subject to $x\in S$. Using the well-known Weierstrasss theorem to get the result.