Epigraphs and lower semicontinuity

The epigraph of f is the set of pairs (x,t) with real height t at least f(x). An infinite value contributes no vertical fiber. The function is convex exactly when this set is convex. In finite-dimensional Euclidean space, f is lower semicontinuous when every sequence converging to x has lower limiting cost at least f(x). Equivalently, its epigraph is closed; equivalently, every finite sublevel set is closed. The sequential formulation remains valid for extended values, including the requirement that costs diverge to infinity near a point assigned infinity.

For the epigraph equivalence, if (x_n,t_n) converges and f(x_n)≤t_n, lower semicontinuity gives f(x)≤liminf f(x_n)≤t. Conversely, a failure of lower semicontinuity supplies a sequence with costs bounded below f(x) by a fixed gap. Passing to a suitable finite height constructs epigraph points converging outside the epigraph. This is why convex analysts call a lower semicontinuous proper convex function closed: the closed object is its epigraph, not necessarily its ordinary graph or its effective domain.

Consider f(x)=1/x for x>0 and infinity otherwise. Its finite part is convex, and it tends to infinity as x approaches zero from the right. Its epigraph is therefore closed even though its domain (0,∞) is not. By contrast, the indicator of the open ray (0,∞) is not lower semicontinuous at zero: points approaching from the right have cost zero, while the assigned value at zero is infinity. Closing that epigraph adds the vertical fiber above zero and yields the indicator of [0,∞).

A less obvious failure occurs when a boundary value is too high but finite. Define h(0)=1, h(x)=0 for x>0, and h(x)=infinity for x<0. This function is convex: mixtures involving zero have right-side cost at least zero. It is not lower semicontinuous at zero because nearby positive points have cost zero. Replacing h(0) by zero closes the epigraph. Checking convexity alone would miss this defect. Closing a proper convex function with an affine lower bound preserves the supporting affine information while repairing such missing limit points.

Let f be proper and lower semicontinuous, with some nonempty compact sublevel set at a finite height. Restrict f to that set and take a minimizing sequence there. Compactness produces a convergent subsequence, and lower semicontinuity gives a minimizer. All points outside that sublevel have larger cost than its defining height, so the minimizer is global. Convexity is not needed for this existence argument, although it adds useful geometry afterward. If all finite sublevels are bounded, lower semicontinuity makes them compact in finite dimensions.

Coercivity means f(x) tends to infinity as the norm of x tends to infinity. For a proper lower semicontinuous function on finite-dimensional space, it provides the bounded sublevels needed for existence. Without coercivity, existence may still hold, but cannot be inferred from closedness alone: exp(x) is continuous convex and has unattained infimum zero as x tends to negative infinity. In an infinite-dimensional Hilbert space, closed bounded sets need not be norm compact; weak compactness and weak lower semicontinuity require a distinct argument rather than reusing the finite-dimensional proof word for word.