Does $\min_{\textbf{x}} \{f(x)| \|\textbf{x}\|_{2}\leq C \}=\min_{\textbf{x}} \{f(x)| \|\textbf{x}\|_{2}=C \}$ hold, if f convex?

Question

For some function $f:V\rightarrow \mathbb{R}$ I've been able to show that it's convex. It is also not hard to show that the set $V=\{\textbf{x}\in\mathbb{R}^{n} | \|\textbf{x}\|_{2}\}$ is a convex set. So, now I'm wondering if the following equality holds: $\min_{\textbf{x}} \{f(x)| \|\textbf{x}\|_{2}\leq C \}=\min_{\textbf{x}} \{f(x)| \|\textbf{x}\|_{2}=C \}$, for some constant $C\in \mathbb{R}$.

I've been able to find theorems regarding the attainment of the maximum of convex functions at the boundary of its convex domain (*), but not a similar theorem for the minimum. A source to a theorem which enables me to justify the above equality would be a really helpful answer.

Thanks in advance!

-Anil

(*): Theorem 3.1 on page 131 of 'Convexity and Optimization in Rn' by Leonard D. Berkovitz; https://books.google.nl/books?id=f6u9NuiCx7AC&pg=PA137&lpg=PA137&dq=convex+function+attains+minimum+at+boundary&source=bl&ots=ImDjVwGWi1&sig=cx7DFbLWr20WnTRW36jFt39abiE&hl=nl&sa=X&ved=0ahUKEwiwjcGowsXOAhUGmBoKHUX-B-4Q6AEIZTAJ#v=onepage&q=convex%20function%20attains%20minimum%20at%20boundary&f=false

Answer 1

No, convex functions do not generally attain their minimum at the boundary. The prototypical example is $f:\Bbb R^2 \to \Bbb R$ with $$ f(x,y) = x^2 + y^2 $$ if we consider $f$ over the unit disk, then it attains its minimum of $0$ at the center, $(0,0)$.

Answer 2

Consider $V = \mathbb{R}$ and $f(x) = |x|$ on $\mathbb{R}$. Then for $C > 0$, $$\min_{\vert x \vert \leq C} f(x) = 0 < C = \min_{\vert x \vert = C} f(x).$$

In a more general setting, just take $f(\mathbf{x}) = \|\mathbf{x}\|_2$ as a counter example.