Let $M$ be a normed space and $A$ a subset (nonempty) of the unit sphere ($S_1$, which is the points of norm $1$). Define, for $\alpha >0$, a $A^{\alpha}=\{x\in S_1 | d\left( x,A\right) \leq\alpha\}$. What can we say about the following properties of $A^\alpha$:
- closedness
- convexity
- $A\subset A^\alpha$
- conic hull of $A^\alpha$ convexity ?
My attempt is in what follows:
- We see that $A^\alpha$ is $f^{-1}(0,\alpha)$, where $f$ is the inverse function of the distance function to the set $A$. Since $f^{-1}$ is continuous, it follows that $A^\alpha$ is closed.
- The set $A^\alpha$ is convex if and only if $A$ is a singleton because because the distance is so.
- $A$ is a subset of $A^\alpha$ from definition.
- I write the definition of conic hull and convexity, but I do not get anything. Thank you for any observation.