I have the following definitions of uniformly convex and strongly convex
Let $f:R^n \to R$ be smooth.
(1) $f$ is uniformly convex if there exists $\theta > 0 $ such that $$\Sigma_{i,j}f_{x_i x_j} \xi_i \xi_j \geq \theta |\xi|^2 \tag1$$ for every $x,\xi \in R^n$.
(2) $f$ is strictly convex if $$\Sigma_{i,j}f_{x_i x_j} \xi_i \xi_j > 0 \tag2$$ for every $x \in R^n$ and every $0 \neq \xi \in R^n $.
I can see that uniformly convex implies strictly convex, but what is an example of a strictly convex function that is not uniformly convex? Also, is there a stronger notion than uniformly convex?
Best Answer
$f(x)=e^x$ is strictly convex but not strongly or uniformly convex. The scalar definition of uniform convexity reduces to $f''(x)>\theta$. For any fixed $\theta>0$, this inequality fails for any $x\le\log(\theta)$.
Indeed, consider any smooth, positive function that approaches zero asymptotically; integrate twice (from, say, the origin) and you have another counterexample.
If you need a multivariate example then just consider $f(x)=\sum_i e^{x_i}.$