In the wild universe of shapes and spaces, not every object stays neatly boxed. Some bend, stretch, or drift toward infinity, and mathematicians have long wrestled with how to talk about them without collapsing into chaos. A recent line of work from Luisa F. Higueras-Montaño at the Universidad Nacional Autónoma de México asks a very human question: when a family of star-shaped sets isn’t bounded or closed, how do we even say that two shapes are “the same” or that one is a little larger than another? The answer, it turns out, is a fresh toolkit that mirrors the everyday intuition we use to compare, say, clouds or breasts of fog in a long-exposure photograph, but in a precise mathematical language. Higueras-Montaño’s work builds radial cousins of classic topologies, introducing new ways to measure distance, talk about convergence, and even talk about dualities that link shapes to their polar counterparts. The study is anchored at UNAM, led by Luisa F. Higueras-Montaño, and it opens a window onto how we can handle the unbounded and the imperfect with the same mathematical grace we use for tidy, bounded objects.
To get a feel for what’s new, imagine you’re charting the set of all possible star-shaped regions in d-dimensional space, where every line from the origin to a point in the shape sits inside the shape up to some radial limit. These “star bodies” might flare out to infinity in some directions, or be non-closed in a way that makes traditional geometry uncomfortable. Higueras-Montaño introduces a radial distance function, a natural analogue of the ordinary distance to a set, but measured along rays emanating from the origin. This is not just a cosmetic change: it reshapes how convergence is defined, how we compare one star body to another, and how robust properties (like dualities) behave when you let shapes wander freely through the spaces they inhabit. The result is a principled way to talk about limits of unbounded star-shaped sets, without demanding they be closed or bounded from the start. It’s geometry that plays well with the messy edges of real-world shapes, not just the clean, static ones we like to draw on a whiteboard.
In short, the paper asks: if we replace the classical distance function with a radial cousin, what kinds of convergence do we get? Do familiar dualities survive unscathed in this radial universe? And what does this tell us about the geometry of the “flowers” that arise when you look at convex bodies through this lens? The answers are surprisingly tidy: two radial topologies emerge—a radial Wijsman type topology and a radial Attouch-Wets type topology. One is not metrizable in this setting, the other is completely metrizable and comes with a concrete radial Attouch-Wets distance. And yes, the duality that maps a shape to its reciprocal in radial terms remains continuous under these new topologies. These technical results may sound abstract, but they matter because they give us a robust, stable framework for discussing shapes that otherwise refuse to behave themselves.
Radial distance as a measuring stick
At the heart of the construction is a simple but powerful idea: define a distance from a point x in Euclidean space to a star body A not by the usual point-to-set distance, but by asking how far you must extend A along the line from the origin through x before it contains x. Put more plainly, think of A as a region that can be probed ray by ray from the origin; for any direction, you record how far along that ray you need to go before you’ve reached x. This quantity is called the radial distance, denoted dr(x, A). It’s a radial, directional cousin of the usual distance to a set, and it has a handful of telling properties that aren’t true for the ordinary distance:
dr(x, A) = 0 exactly when x is in A, which makes the function a faithful, ray-by-ray fingerprint of the shape. If x lies outside A, dr(x, A) records how far you must go along the ray θx to reach the boundary of A in that direction. The radial distance is defined in terms of something Higueras-Montaño calls a radial sum, a way of adding star bodies by adding their radial functions. This is not just a quirky definition; it’s what makes dr a reliable, injective map from star bodies to function space. In particular, if you know dr(·, A) for all points x in Rd, you can recover A exactly.
One striking and useful feature is that dr can be discontinuous in general. That’s a hint that radial geometry lives on a different landscape from the classic, nicely behaved closed sets we learn in elementary geometry. Yet when A has a continuous radial function—a condition the author denotes Sd1—dr(·, A) becomes continuous, and the star body is then closed. This dichotomy helps explain why the radial topologies behave differently from their classical cousins: some edge cases you can ignore in the bounded world suddenly become central when unboundedness is allowed.
Importantly, the radial distance isn’t just a curiosity; it’s a tool. The author shows, for example, that the radial distance functional completely controls a variant of the radial metric δ between two bounded star bodies. In other words, dr provides a concrete, computable way to measure how far apart two star shapes are direction by direction, and the resulting numbers bridge to the usual radial metric in a natural way. This is the seed from which the whole topological construction grows.
Toward a radial language for convergence
Convergence is the central notion that lets us talk about sequences of shapes approaching a limit. In classic set-valued topology, Wijsman convergence and Attouch-Wets convergence are two characterizations of how shapes can converge when distance notions exist. Higueras-Montaño defines radial versions of these two notions. The radial Wijsman topology, τW_r, is the weakest topology that makes every dr(x, ·) a continuous function of the shape. In practical terms, a net (a fancy generalization of a sequence) of star bodies Ai in Sd_rc converges to A in τW_r if and only if the radial distances dr(x, Ai) converge pointwise to dr(x, A) for every x in Rd.
The punchline here is twofold. First, τW_r is not metrizable on the whole Sd_rc, even when you restrict to the nicely-behaved subset Sd1. That means you can have convergent nets that refuse to be knocked into a metric space by a single, countably additive distance. Second, and more cheerfully, Higueras-Montaño introduces a radial Attouch-Wets topology, τAW_r, which is completely metrizable. In this setup, there exists a well-behaved radial Attouch-Wets distance dAW_r that generates τAW_r, turning the space Sd_rc into a complete metric space with a robust notion of convergence that plays nicely with both bounded and unbounded stars. The comparison with the classical Attouch-Wets distance dAW is clean: for any pair of closed star bodies A1, A2, one always has dAW(A1, A2) ≤ dAW_r(A1, A2). This is a comforting inequality because it ties the new radial world back to the familiar, well-trodden terrain of convex-geometric analysis.
In more down-to-earth terms, τW_r tells you how a sequence looks when you stare down every ray from the origin and compare radial distances term-by-term. τAW_r, by contrast, adds a uniform-taster, watching how those radial distances behave uniformly on every bounded chunk of Rd. The upshot is a powerful pair of convergence notions: one that captures pointwise radial behavior, and another that captures uniform radial behavior on bounded regions. That duo is what makes the radial theory both flexible and sturdy, able to handle the wilder shapes that skip the usual compactness assumptions.
A distance that feels like a map radial Attouch-Wets style
To translate these convergence ideas into something you can compute and compare, Higueras-Montaño defines a concrete radial Attouch-Wets distance, dAW_r. The definition looks technical, but the spirit is simple: you measure how far two star bodies are by looking at how their radial distances differ on growing balls around the origin. Concretely, you take a supremum over all radii j, but you cap each term by 1/j to ensure the distance remains finite and manageable. Inside, you look at the maximum discrepancy between dr(x, A1) and dr(x, A2) for all x with norm up to j, then you dampen the influence of large j with the 1/j cap. This yields a metric that is complete, compatible with the topology τAW_r, and, crucially, aligns with the intuitive notion that two shapes should look the same where you can actually observe them up close and in bounded regions.
One of the paper’s neat results is that dAW_r dominates the classical Attouch-Wets distance dAW when you compare two closed star bodies. In practical terms: radial observations never overstate how far apart shapes are in the full, global sense. They only understate, never overstate, the separation, which makes the radial distance a reliable, conservative tool for analysis. On the other hand, on the subfamily of compact, origin-containing convex bodies (the Sd1,(0),b world), the radial Attouch-Wets topology and the old Attouch-Wets distance align in the sense that the same topology is generated by the two measures. In short: the radial framework fits nicely with the classical one where the shapes behave well, while still giving you extra leverage when you let shapes roam to infinity.
Beyond its mathematical neatness, this radial distance works as a sticky lens for comparing shapes sliced by bounded regions. A key technical lemma shows that you can compute dAW_r by looking at the radial distance in ever-smaller outer shells and then stitching those observations together. In the authors’ language, dr and δ—the radial metric—cohere beautifully with dAW_r on the relevant subfamilies. The upshot is a toolbox in which you can talk about convergence with both local precision and global perspective, depending on what your application demands.
Continuity of star duality and the flowers
One of the paper’s central notions is a duality, denoted Φ, that acts on star-shaped sets by inverting their radial profile. In intuitive terms, Φ turns a shape into something like a reciprocal in radial coordinates: where the original shape extends far in a direction, Φ compresses that extension, and vice versa. The nice surprise is that this duality remains continuous when you measure shapes with the radial Wijsman or radial Attouch-Wets topologies. The author proves that Φ is a continuous duality on Sd_rc with respect to τW_r and τAW_r, and even its restriction to Sd1 inherits continuity with respect to the radial Attouch-Wets metric dAW_r. This is not a trivial statement: familiar polar duality is delicate under topological changes, but the radial framework preserves its stability.
Connected to Φ is the geometric notion of a flower, which Milman, Milman and Rotem introduced to capture a hidden duality structure in convex geometry. Flowers encode, in a single star body, a whole family of support-like data from convex bodies. Higueras-Montaño shows that the duality Φ we just discussed can be interpreted through flowers, tying together two threads of a larger geometric story: how dualities show up when you repackage shape data as radial functions, and how those dualities behave under radial topologies. The work even defines a playful distance on the family of flowers and explores how this distance interacts with Φ, revealing both the promises and the subtle pitfalls of this approach. The upshot is a more unified picture of dualities in convex and star-shaped geometry, one that survives the passage to unbounded, non-closed worlds.
Perhaps most striking is the practical consequence: the star duality Φ acts as a bridge between the earthier concrete world of star bodies and the more algebraic, polar world of convex sets. On the one hand, the classical polar duality is central to many areas of convex analysis; on the other hand, the radial translation via Φ allows us to move back and forth with continuity guarantees under the radial topologies. This isn’t just a mathematical curiosity. It’s a conceptual tool for understanding how dual representations of shapes behave in spaces that stretch without bound, which matters in optimization, geometric tomography, and computational geometry where unbounded shapes crop up frequently.
Why this matters beyond math
The technical triumphs in Higueras-Montaño’s paper are grounded in a bigger narrative: when you loosen the shackles of boundedness and closedness, you don’t have to abandon structure; you can redesign the structure so that it captures what you really care about—how shapes relate to one another, how they converge, and how dual shapes reflect those relationships. This radial framework equips researchers with a robust way to talk about convergence of unbounded star-shaped sets, which pop up in a surprising number of places, from abstract functional analysis to computational geometry and even data science where objects are often measured by their extents along many directions rather than by a tidy boundary alone.
For a field built on precise definitions and careful limits, the practical upshot is a sturdier way to talk about “shape space.” If you’re modeling a family of regions that can grow without bound, or comparing noisy data-derived shapes that aren’t neatly closed, the radial topologies give you a principled language to discuss how those shapes stabilize (or fail to stabilize) under perturbations. And when you care about dual descriptions—how one shape encodes the information of another—the radial versions of Wijsman and Attouch-Wets topologies ensure those dualities behave continuously as you vary the shapes. This is the kind of mathematical maturity that often quietly underwrites advances in adjacent fields: algorithms for shape matching, numerical optimization in high dimensions, or even new ways to reason about data geometry where the data live in unbounded feature spaces.
The work is anchored in the mathematical program around star-shaped geometry and duality, but its implications ripple outward. It clarifies how to compare irregular shapes in a stable way, even when those shapes don’t fit into neat boxes. It also offers concrete new tools—the radial distance functionals, the radial Attouch-Wets distance, and the radial Wijsman topology—that other researchers can adapt to their own problems. It’s not a flashy breakthrough in the sense of a laboratory experiment or a single dramatic discovery; rather, it’s a careful expansion of the mathematical vocabulary that lets us talk about the geometry of infinity with sharper arrows in our quiver.
In the end, Higueras-Montaño’s radial-topology program gives us a clearer map of a geography that was already there, but was hard to navigate. It invites us to reimagine the shapes we study—not as rigid blocks bound by borders, but as flexible, directionally rich objects whose most meaningful comparisons come from looking along every ray emanating from the origin. The work is a reminder that in mathematics, as in life, the way you frame the problem can liberate you from the old constraints and illuminate the path forward. And it’s heartening to know that the journey is being charted at a venerable institution like UNAM, with a thoughtful voice leading the way into new vistas of convergence, duality, and shape-aware geometry.
