Example of a uniformly convex domain in $\Bbb R^n$

Question

I am trying to understand the differences between a convex domain, and a uniformly convex domain.

Intuitively, to my knowledge, a convex domain is one where any line between any two points in the domain lies inside the domain, so an example to my mind would be $[0,1]\times[0,1]\subset\Bbb R^2$.

Can someone give a basic example of a uniformly convex domain in $\Bbb R^n$ (or $\Bbb R^2$)?

Are there relations to do with compactness?

Thanks all!

Answer 1

Consider the closed ball $\{x: ||x||\le 1\}$, where $||\cdot ||$ is the Euclidean norm on $R^n$.

Edit: Here are the promised details. Recall that a subset $X$ of a vector space $V$ is convex if for every $x,y\in X$, the segment $xy$ is contained in $V$. A (closed) subset $X$ is strictly convex whenever for distinct $x,y\in X$ their midpoint $(x+y)/2$ is contained in the interior of $X$. The notion of uniform convexity is a quantification of strict convexity in the case when $V$ is a normed vector space (say, a Euclidean space $R^n$): One requires that for every $\epsilon>0$ there exists $\delta>0$ such that if $x,y\in X$ and $||x-y||\ge \epsilon$ then the distance from $(x+y)/2$ to the boundary of $X$ is at least $\delta$. Thus, the relation between uniform convexity and convexity is akin to the relation of continuity and uniform continuity. (Except the inequalities go in the opposite direction.)

To understand what uniform convexity means, consider its negation: If $X$ is not uniformly convex, then there exists $\epsilon>0$ and a sequence of points $x_i, y_i\in X$ such that $||x_i-y_i||>\epsilon$ while the distance from $(x_i+y_i)/2$ to the boundary of $X$ converges to zero as $i\to\infty$. Now, if $X$ is also compact, then, after passing to a subsequence and taking a limit, we conclude that such $X$ is not strictly convex. Therefore, in the case of compact sets, uniform convexity is equivalent to strict convexity. (Just like for uniform continuity and continuity!) If $X$ is noncompact, then strict convexity does not imply uniform convexity (consider the epigraph of the function $y=e^x$ in the plane).