Beyond the Curve: Unpacking ω-Convexity and Its Hahn-Banach Secrets

You know, in the world of math, especially when we’re trying to figure out how to optimize things or understand shapes and structures in abstract spaces, convexity is a real superstar. Think of a simple bowl shape – if you pick any two points inside, the line connecting them stays *inside* the bowl. That’s the essence of convexity, and it leads to all sorts of beautiful and useful properties.

What’s the Big Deal About Convexity?

Standard convex functions are incredibly well-behaved. They have nice properties like continuity under certain conditions, and they’re fundamental in areas like optimization theory and functional analysis. If you’ve ever tried to find the minimum of a function, you know how much easier it is if that function is convex!

We often define convexity using something called an ‘affine difference’. Basically, it measures how much a function deviates from being a straight line between two points. If that deviation is always non-positive, the function is convex. If it’s non-negative, it’s concave. If it’s zero, it’s affine (a straight line or plane).

Stepping Beyond: Generalized Convexity

But sometimes, the real world, or even just more complex mathematical problems, don’t fit neatly into the standard convex box. Functions might be *almost* convex, or convex in a slightly different way. This is where the idea of generalized convexity comes in. Mathematicians have been playing with this idea for a while, trying to modify the definition of convexity to capture a wider range of functions while still keeping some of those sweet, sweet properties.

For instance, people have looked at ε-convex functions (where the deviation from linearity is bounded by a small ε) or strongly convex functions (where the deviation is bounded *below* by a term related to the distance between the points, making them ‘more’ convex). These generalizations have proven super useful in areas like optimal control and optimization.

Enter ω-Convexity

The paper we’re diving into introduces a really broad way to generalize convexity: ω-convexity. Instead of just adding a constant (like ε) or a specific distance-based term, they introduce a general function, ω (omega), that depends on the two points you pick and the point between them.

A function f is called ω-convex if, for any two points x and y in its domain and any point tx + (1-t)y on the line segment between them (where t is between 0 and 1), the value of the function at the intermediate point is less than or equal to the weighted average of the function values at x and y *plus* this ω term:
f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) + ω(x, y, t)

See? It’s the classic convexity inequality, just with that extra ω(x, y, t) bit thrown in. If ω is always zero, you get standard convexity back. If ω is ε, you get ε-convexity. This framework is really flexible and covers many other types of generalized convexity you might find in the literature.

First Steps: Basic Properties

So, once you have this new definition, the first thing you do is check if the nice properties of standard convex functions still hold, perhaps in a modified way. The paper does just that.

They show that ω-convex functions behave nicely under certain operations. For example:

• If you have a whole bunch of ω-convex functions, their supremum (the smallest function that’s greater than or equal to all of them pointwise) is also ω-convex.
• If you have a ‘chain’ of ω-convex functions (meaning for any two functions in the set, one is always less than or equal to the other), their infimum (the largest function that’s less than or equal to all of them) is also ω-convex.

These are pretty standard results for various types of convex functions, and it’s good to see they extend here.

They also look at the epigraph of an ω-convex function. The epigraph is just the set of points ‘on or above’ the function’s graph. For standard convex functions, the epigraph is always a convex set. For ω-convex functions, they find a similar characterization involving the ω term. It’s like the shape ‘below’ the function still tells you a lot about its ω-convex nature.

And yes, they prove a version of the famous Jensen inequality for ω-convex functions. The standard Jensen inequality relates the function of an average to the average of the function values for convex functions. For ω-convex functions, you get a similar inequality, but – you guessed it – with the ω term making an appearance. It’s the classic idea extended:
f(t₁x₁ + … + tnxn) ≤ t₁f(x₁) + … + tn f(xn) + Σ ω terms… (The exact form involves sums of ω terms, making it a bit more complex than the simple definition, but the spirit is the same).

Smooth Moves: Continuity

In math, especially when we’re dealing with spaces that have a notion of ‘closeness’ (like topological spaces), we really care about continuity. A continuous function is one you can draw without lifting your pen – no sudden jumps. For standard convex functions, local boundedness above at an interior point is enough to guarantee continuity.

The paper investigates when ω-convex functions are continuous. It turns out that if the function is defined on a nice space (a real linear topological space), is ω-convex, is locally bounded above at an interior point, *and* the ω function satisfies a certain condition related to approaching zero as you move between points, then the function is continuous at that point. This is a powerful result because ‘locally bounded above’ is a much weaker condition than assuming continuity everywhere. They also show that upper semicontinuity combined with conditions on ω can imply continuity.

They also explore local boundedness itself. If an ω-convex function is locally bounded above at one point in an open convex set, and the ω function satisfies another local boundedness condition, then the function is locally bounded above everywhere in that set.

For functions defined on a Baire space (another type of space with useful properties), they show that a lower semicontinuous ω-convex function is continuous at any ‘algebraically internal’ point (a concept related to interior points) provided ω satisfies certain conditions. It’s pretty neat how these abstract conditions on ω and the space itself can force a function to be continuous.

The Main Event: Hahn-Banach for ω-Convexity

Now, let’s talk about the big guns: the Hahn-Banach theorems. These are absolutely fundamental results in functional analysis. In their classic forms, they basically say you can extend a linear function defined on a subspace to the whole space while preserving its norm (extension form), or that you can separate two disjoint convex sets with a hyperplane (separation form). They guarantee the existence of linear functionals with specific properties.

Mathematicians have generalized Hahn-Banach in countless ways. This paper presents Hahn-Banach type theorems for ω-convex functions. The core idea here is finding an ω-affine function that acts like a ‘support’ or an ‘extension’ for an ω-convex function under certain conditions.

Think of a convex function and its tangent line at a point. The tangent line ‘supports’ the function from below. A Hahn-Banach type theorem in this context might say that under certain conditions, you can find an ω-affine function that supports your ω-convex function in a similar way.

The paper proves both a topological version (for continuous functions on topological spaces) and an algebraic version (on just a real linear space, without needing continuity initially). The key results state that if you have a continuous ω-convex function defined on a convex, open set, and it’s ω-affine on a smaller convex subset with a specific property related to ‘extreme points’ (basically, points that can’t be written as a non-trivial convex combination of other points in the set), then you can extend this ω-affine behavior to an ω-affine function on the whole set *if and only if* the ω function satisfies two specific conditions (labeled ‘a’ and ‘b’ in the paper). The algebraic version is similar but doesn’t require continuity assumptions upfront.

Support and Representation

These Hahn-Banach type theorems immediately lead to other cool results, like support theorems. A support theorem for ω-convex functions says that under the right conditions (including those conditions ‘a’ and ‘b’ on ω), for any point in the domain of a continuous ω-convex function, you can find a continuous ω-affine function that ‘supports’ it at that point (meaning the ω-affine function is less than or equal to the ω-convex function everywhere and equal at that specific point). This is like finding a generalized tangent function.

Support theorems are super important because they often lead to representation theorems. For standard convex functions, the support theorem implies that a convex function can be represented as the pointwise supremum (the upper envelope) of all the affine functions that support it. This paper shows a similar result for continuous ω-convex functions: they can be represented as the pointwise supremum of continuous ω-affine functions that support them, provided ω satisfies conditions ‘a’ and ‘b’. This is a powerful way to understand the structure of these generalized convex functions.

A Twist in the Tale: Existence Issues

Here’s a bit of a curveball. Even if the ω function satisfies the nice conditions (‘a’ and ‘b’) that make the Hahn-Banach and support theorems work, it turns out you might not even *find* any ω-convex functions at all! The paper includes a theorem showing that for certain types of ω functions (specifically, one related to the norm in a normed space) on a convex set that isn’t just a single point, no function can satisfy the ω-convexity inequality. This is a fascinating result – the framework is general, the theorems are powerful *if* an ω-convex function exists, but existence isn’t guaranteed just by the properties of ω.

Wrapping Up

So, what’s the big picture? This paper takes the fundamental idea of convexity and stretches it, creating the concept of ω-convexity, which is flexible enough to cover many other generalizations. By carefully studying the properties of these ω-convex functions, particularly their behavior under operations and conditions for continuity, the authors build up to the main event: proving Hahn-Banach type theorems. These theorems are crucial because they provide tools to understand the structure of ω-convex functions, showing when they can be supported or represented by simpler ω-affine functions, provided the ‘wiggle room’ function ω behaves in a specific way. The surprising non-existence result for certain ω functions adds another layer of depth to the theory. It’s pretty exciting stuff, pushing the boundaries of classic mathematical ideas and providing new tools for tackling complex problems in analysis and optimization.

Source: Springer