Visual help to understand epi-convergence

Question

I have read the definition of epi-convergence in wikipedia, but would like to get a sense of definition by some visual example (or at least easy to understand functions). Any help?

Answer 1

As I wrote elsewhere, it helps to think in terms of epigraphs $\operatorname{ep} f = \{(x,y)\colon y\ge f(x)\}$. Epiconvergence is precisely the Kuratowski convergence of epigraphs. The epi-convergence $f_k\to f$ can be visualized as:

• every point $p\in \operatorname{ep}f$ is being approached by the epigraphs $\operatorname{ep} f_k$ in the sense that $\operatorname{dist}(p, \operatorname{ep}f_k)\to 0$
• every other point $p\notin \operatorname{ep}f$ stays away from the epigraphs $\operatorname{ep} f_k$ in the sense that $\liminf \operatorname{dist}(p, \operatorname{ep}f_k) > 0$.

The simplest example is the constant sequence $f_1=f_2=\dots$. The Kuratowski limit is then simply the closure of the set, so $\operatorname{ep} f =\overline{\operatorname{ep}f_1}$, which means $f$ is the lower semicontinuous regularization of $f_1$.

For a nontrivial example, consider $f_k(x)=-\exp(-kx^2)$. The epigraphs are the upper half-plane with a narrow downward bump attached near $0$. The width of this bump tends to zero, and in the limit it becomes the line segment from $(0,0)$ to $(0,-1)$. We get the epigraph of the function $f$ such that $f(0)=-1$, and $f(x)=0$ otherwise.

Also consider $f_k(x)= \exp(-kx^2)$. Now the epigraphs consist the upper half-plane with a narrow upward cavity near $0$. The width of the cavity tends to zero, and in the limit it disappears. We get the epigraph of the constant function $f\equiv 0$.

The Kuratowski limit is always closed, so the epi-limit is always a lower semicontinuous function.

Finally, $f_n(x) = (-1)^n \exp(-nx^2)$ is a sequence that fails to epi-converge: the lower and upper Kuratowski limits are different.