Convex Set - Sarcasm Wiki

Contents

1. Overview
2. Etymology
3. Cultural Impact

Oh, this again. Very well. If you insist on delving into the foundational minutiae of geometry , let’s discuss “convex sets.” Don’t expect fireworks; this is less about revelation and more about the tedious precision required to build anything remotely stable in mathematics.

In geometry , a set of points is designated as convex if, and only if, for any two arbitrary points chosen from within that set, the entire line segment connecting those two points also lies completely within the confines of the set. This isn’t just a casual observation; it’s a definitive characteristic, a stringent requirement for a set to earn the label “convex.” You can think of it as a spatial integrity test: if you can draw a straight path between any two internal points and ever find yourself outside the set, then it simply fails.

Consider, for instance, a solid cube . Pick any two points inside it, or on its surface, and the line connecting them will, without fail, remain entirely within the cube. Ergo, a solid cube is a prime example of a convex set. Now, contrast that with something distinctly not convex. Imagine a crescent shape, perhaps the slender sliver of a new moon. If you select two points on the tips of that crescent, the line segment joining them would undoubtedly slice through the empty space outside the crescent’s material. Thus, anything that is hollow, or possesses an indentation—such as that crescent shape, or a simple letter ‘C’—is unequivocally not convex. It’s a rather straightforward, if demanding, distinction.

The boundary of a convex set, particularly when observed in a two-dimensional plane, consistently manifests as a convex curve . This isn’t a coincidence but a direct topological consequence of the set’s inherent convexity. Furthermore, if you take any arbitrary subset ‘A’ within a Euclidean space , the intersection of all possible convex sets that fully contain ‘A’ yields a uniquely defined entity known as the convex hull of ‘A’. This ‘convex hull’ is, by definition, the smallest possible convex set that can completely enclose the original, potentially non-convex, subset ‘A’. It’s like finding the tightest, most efficient convex wrapper for whatever chaotic collection of points you initially present.

Moving beyond mere spatial arrangements, the concept of convexity permeates functional analysis. A convex function is formally defined as a real-valued function operating over a specific interval on the real line, characterized by a crucial property: its epigraph —which is the set of all points lying on or above the graph of the function—must itself constitute a convex set. This seemingly abstract definition is incredibly powerful. The field of convex minimization , a significant subfield within mathematical optimization , is dedicated precisely to solving problems that involve finding the minimum values of these convex functions, constrained within convex sets. The broader academic discipline entirely devoted to meticulously studying the intricate properties of both convex sets and convex functions is known as convex analysis . It’s a vast, interconnected web of ideas, all stemming from this fundamental concept.

The spaces in which these elegant convex sets are most commonly defined and rigorously studied include the familiar Euclidean spaces , the more abstract affine spaces over the field of real numbers , and even extends to certain sophisticated non-Euclidean geometries . The ubiquity of this definition underscores its fundamental importance across diverse mathematical landscapes.

Definitions

Ah, the definitions. The bedrock, the unyielding truth. Pay attention, because precision matters here more than your feelings.

Let S be a vector space or an affine space operating over the field of real numbers , or, for a more expansive view, over some ordered field . This general framework, naturally, encompasses the specific instance of Euclidean spaces , which are themselves a particular type of affine space. A subset C of this space S is formally declared convex if, for every single pair of points x and y that belong to C, the entire line segment connecting x and y is also unequivocally included within C. This is not a suggestion; it is a strict, non-negotiable condition.

What this fundamentally means is that any affine combination of the form (1 − t ) x + ty must belong to C for all x, y in C and for any scalar t within the closed interval [0, 1]. This formulation is not just a mathematical convenience; it profoundly implies that the property of convexity remains absolutely invariant under any affine transformation . You can stretch, skew, or translate a convex set, and it will remain convex. Furthermore, this definition carries a significant topological consequence: a convex set situated within a real or complex topological vector space is inherently path-connected , which, as a direct corollary, means it is also connected . One might even call it “well-behaved,” if one were prone to such sentimentalities.

To introduce a further layer of refinement, a set C is termed strictly convex if every single point on the line segment connecting x and y—with the sole exception of the endpoints themselves—resides strictly within the topological interior of C. This is a more demanding criterion than simple convexity. A closed convex subset is strictly convex if and only if every one of its boundary points qualifies as an extreme point – a point that cannot be expressed as a convex combination of any two other distinct points in the set. It’s about the sharp edges, or lack thereof.

And for those who appreciate even more structure, a set C is defined as absolutely convex if it simultaneously satisfies two conditions: it must be convex, and it must also be balanced . Such sets possess a particular symmetry around the origin, adding another dimension to their ordered nature.

Examples

One would hope these are self-evident, but let’s humor the necessity of illustration.

The convex subsets of R, which is simply the set of all real numbers , are quite elementary: they consist of all possible intervals (open, closed, or half-open) and, trivially, individual points within R. In the more expansive realm of the Euclidean plane (R²), some archetypal examples of convex subsets include solid regular polygons —meaning the polygon itself and its entire enclosed region—solid triangles, and any intersection resulting from multiple solid triangles. These are the shapes that don’t have unexpected indentations or voids. Extending into a Euclidean 3-dimensional space (R³), canonical convex subsets encompass the well-known Archimedean solids and the perfectly symmetrical Platonic solids . These are the geometric ideals, the shapes that perfectly contain all their internal line segments. On the other side of the coin, examples of sets that decidedly fail the convexity test are the intricate and often self-intersecting Kepler-Poinsot polyhedra ; their complex structures prevent them from satisfying the fundamental definition.

Non-convex set

A set that, by its very nature, does not conform to the strict definition of convexity is, rather unimaginatively, referred to as a non-convex set. In the specific context of polygons, a polygon that fails to be a convex polygon is occasionally, and somewhat colloquially, termed a concave polygon . Some sources, perhaps out of a misguided desire for symmetry in terminology, might more generally employ the term “concave set” to denote any non-convex set. However, let’s be clear: most authoritative mathematical sources emphatically prohibit this usage. It’s an imprecise and misleading term, as concave function has a precise definition, and conflating sets with functions is simply sloppy. There is no such thing as a “concave set” in rigorous mathematical discourse; one simply refers to it as a non-convex set.

Interestingly, the complement of a convex set—for instance, the region below the graph of a concave function , which is essentially the complement of its epigraph —is sometimes given the specific designation of a reverse convex set. This terminology is particularly prevalent and useful within the specialized domain of mathematical optimization , where such sets often represent constraints that lead to more complex, non-convex problems. It’s a way of acknowledging their relationship to convexity without mislabeling them.

Properties

Now, for some of the more universal truths governing these sets.

Given an arbitrary number, say r, of points—u₁, …, uᵣ—all residing within a convex set S, and given r corresponding nonnegative numbers —λ₁, …, λᵣ—such that their sum precisely equals 1 (i.e., λ₁ + … + λᵣ = 1), then the resultant affine combination expressed as:

$${\displaystyle \sum _{k=1}^{r}\lambda {k}u{k}}$$

will, without exception, also belong to S. This is a powerful generalization. Considering that the foundational definition of a convex set is inherently the case where r = 2, this property effectively serves to fully characterize convex sets for any finite number of points. It’s a fundamental extension of the basic principle.

Such an affine combination, with the non-negative coefficients summing to one, is specifically termed a convex combination of the points u₁, …, uᵣ. Building on this, the convex hull of any subset S within a real vector space is precisely defined as the intersection of all possible convex sets that fully encompass S. More concretely, and perhaps more intuitively, the convex hull is simply the complete collection of all possible convex combinations that can be formed from the points within S. Crucially, and by its very construction, this convex hull is itself always a convex set. It’s the most compact, convex container for any given collection of points.

A (bounded) convex polytope is a specific and important instance of a convex set, defined as the convex hull of a finite collection of points within some Euclidean space Rⁿ. These are the “solid” versions of polygons and polyhedra.

Intersections and unions

The behavior of convex subsets under standard set operations reveals some elegant, if predictable, rules. The collection of all convex subsets residing within a vector space, an affine space, or a Euclidean space possesses the following foundational properties:

The empty set , a set containing absolutely nothing, is trivially convex. Similarly, the entire space itself, encompassing everything, is also considered convex. These are the boundary cases, the null and the universal.
The intersection of any collection of convex sets—whether finite, countable, or uncountable—will invariably result in a set that is also convex. This property is remarkably robust and highly useful in proofs and constructions. If two convex shapes overlap, their common region is also convex.
The union of a collection of convex sets is guaranteed to be convex only if those sets form what is known as a chain under inclusion. This means that for any two sets in the collection, one must be a subset of the other. The restriction to chains here is absolutely critical and cannot be overlooked. The union of two arbitrarily chosen convex sets, which do not necessarily form a chain, is not generally convex. One only needs to picture two disjoint convex shapes to realize their union is unlikely to be convex; for example, two separate solid circles. So, while intersections are always well-behaved, unions demand a more structured relationship.

Closed convex sets

Closed convex sets represent a particularly important class within the family of convex sets. They are defined as convex sets that possess the additional property of containing all their limit points . This means that if you have a sequence of points within the set that converges to some point, that limit point must also be part of the set. These sets can be elegantly characterized as the intersections of closed half-spaces —which are essentially sets of points in space that lie on, and to one side of, a hyperplane .

From the preceding discussion, it’s immediately obvious that any such intersection will not only be convex but will also inherently be a closed set. The more profound and challenging part is proving the converse: that every single closed convex set can, in fact, be represented as such an intersection of closed half-spaces. To establish this converse, one must invoke the powerful supporting hyperplane theorem . This theorem states, in essence, that for any given closed convex set C and any point P located strictly outside of it, there exists at least one closed half-space H that contains C but pointedly excludes P. This theorem is not a trivial result; it stands as a special, yet fundamental, case of the much broader and more abstract Hahn–Banach theorem from the field of functional analysis . It’s a cornerstone for understanding the structure of these sets.

Face of a convex set

A ‘face’ of a convex set, denoted as $F$ within a set $C$, is a concept that delves into the substructure of convexity. Formally, a face $F$ of a convex set $C$ is itself a convex subset of $C$. Its defining characteristic is that whenever a point $p$ in $F$ lies strictly between two points $x$ and $y$ (meaning $p$ is an interior point of the segment $xy$) that are both within the larger set $C$, then both $x$ and $y$ must also be contained within $F$. This is a stringent condition.

Equivalently, and perhaps more algebraically, for any $x, y \in C$ and any real number $t$ such that $0 < t < 1$, if the affine combination $(1-t)x + ty$ happens to be in $F$, then it is absolutely required that both $x$ and $y$ must also be in $F$. This definition ensures that faces are “flat” or “straight” parts of the boundary. According to this precise definition, the set $C$ itself and the empty set are technically considered faces of $C$; these are sometimes, rather predictably, called the trivial faces of $C$. More significantly, an extreme point of $C$ is simply a point that, when considered as a singleton set, forms a face of $C$. These are the “corners” or “vertices” in intuitive terms.

Let’s consider a convex set $C$ in $\mathbb{R}^n$ that is compact —which, in finite-dimensional Euclidean space, is equivalent to being both closed and bounded . A profound result states that such a set $C$ is precisely the convex hull of its extreme points. This means that the entire set can be generated by taking all possible convex combinations of its extreme points. More generally, and more abstractly, each compact convex set within a locally convex topological vector space is the closed convex hull of its extreme points—a truth enshrined in the powerful Krein–Milman theorem . It’s a beautiful distillation of complexity to its fundamental components.

For practical illustration:

A triangle in the plane, including the entire region it encloses, constitutes a compact convex set. Its non-trivial faces are readily identifiable as the three vertices (which are also its extreme points) and the three edges (which are line segments connecting two extreme points). The extreme points, in this case, are exclusively its three vertices.
In contrast, consider the closed unit disk , formally expressed as ${(x,y)\in \mathbb{R}^{2}:x^{2}+y^{2}\leq 1}$. For this set, the only non-trivial faces are its extreme points, which comprise all the points lying directly on the unit circle , $S^{1}={(x,y)\in \mathbb{R}^{2}:x^{2}+y^{2}=1}$. Every point on the boundary of the disk is an extreme point because it cannot be expressed as a strict convex combination of two other distinct points within the disk.

Convex sets and rectangles

Sometimes, even abstract geometry yields to more tangible, if still precise, observations about shape and area. Let $C$ be a convex body in the plane—meaning a convex set whose interior is definitively non-empty. It is a known geometric fact that we can always inscribe a rectangle, let’s call it $r$, within $C$. Furthermore, it is possible to circumscribe a homothetic copy, $R$, of that inscribed rectangle $r$ about $C$. The positive homothety ratio (the scaling factor between $r$ and $R$) is elegantly bounded, always being at most 2. This relationship extends to their areas, establishing a clear inequality:

$${\displaystyle {\tfrac {1}{2}}\cdot \operatorname {Area} (R)\leq \operatorname {Area} (C)\leq 2\cdot \operatorname {Area} (r)}$$

This inequality quantifies a fundamental relationship between a convex body and its rectangular approximations, providing bounds that are surprisingly tight. It’s a nice, neat little truth in a world full of approximations.

Blaschke-Santaló diagrams

For those who enjoy a more sophisticated classification, the realm of planar convex bodies offers structured insights. The set of all planar convex bodies, denoted as ${\mathcal {K}}^{2}$, can be meticulously parameterized. This parameterization typically occurs in terms of three key geometric invariants: the convex body’s diameter $D$ (the largest distance between any two points in the set), its inradius $r$ (the radius of the largest circle that can be entirely contained within the convex body), and its circumradius $R$ (the radius of the smallest circle that can completely enclose the convex body).

In fact, the precise relationships governing these parameters are captured by a specific set of inequalities:

$${\displaystyle 2r\leq D\leq 2R}$$ $${\displaystyle R\leq {\frac {\sqrt {3}}{3}}D}$$ $${\displaystyle r+R\leq D}$$ $${\displaystyle D^{2}{\sqrt {4R^{2}-D^{2}}}\leq 2R(2R+{\sqrt {4R^{2}-D^{2}}})}$$

These inequalities define the permissible region for these parameters. This entire set of relationships can be visualized as the image of a function $g$ that maps any given convex body to a specific point in $\mathbb{R}^2$, represented by the coordinates $(r/R, D/2R)$. The resulting graphical representation of this image is famously known as a (r, D, R) Blaschke-Santaló diagram. This diagram provides a comprehensive visual catalog of all possible planar convex shapes based on these intrinsic properties.

(Insert diagram here, as it was in the original) Blaschke-Santaló (r, D, R) diagram for planar convex bodies.

The diagram often highlights specific, iconic shapes: $\mathbb{L}$ typically denotes the line segment, $\mathbb{I}{\frac {\pi }{3}}$ represents the equilateral triangle, $\mathbb{RT}$ signifies the Reuleaux triangle (a curve of constant width), and $\mathbb{B}{2}$ stands for the unit circle. These points represent the extreme cases and benchmarks within the diagram.

Alternatively, the set ${\mathcal {K}}^{2}$ can also be parameterized using a different, but equally informative, trio of invariants: its width (defined as the smallest distance between any two distinct parallel support hyperplanes), its perimeter, and its area. Each parameterization offers a unique lens through which to analyze and classify the vast diversity of convex forms.

Other properties

Beyond the core definitions and specific examples, convex sets exhibit a suite of other properties, particularly within the framework of topological vector spaces . Let $X$ be a topological vector space, and let $C \subseteq X$ be a convex set. The following properties hold true:

The closure of $C$, denoted $\operatorname{Cl} C$, and the interior of $C$, denoted $\operatorname{Int} C$, are both themselves convex sets. This demonstrates a certain robustness: the “fuzziness” of the boundary or the “emptiness” of the interior doesn’t destroy the underlying convexity.
If $a$ is a point strictly within the interior of $C$ (i.e., $a \in \operatorname{Int} C$) and $b$ is a point on the boundary or within the closure of $C$ (i.e., $b \in \operatorname{Cl} C$), then the “half-open” line segment connecting $a$ to $b$, denoted $[a,b[$, is entirely contained within the interior of $C$. Here, $[a,b[ ,:=\left{(1-r)a+rb:0\leq r<1\right}$. This means that if you start inside and move towards the boundary (or even beyond it, but still within the closure), you remain in the interior until the very last point.
If the interior of $C$ is not empty (i.e., $\operatorname{Int} C \neq \emptyset$), then some powerful relationships emerge:
- The closure of the interior of $C$ is identical to the closure of $C$ itself: $\operatorname{cl} \left(\operatorname{Int} C\right)=\operatorname{Cl} C$. This implies that a non-empty interior “fills out” the entire set up to its boundary.
- The interior of $C$ is equivalent to the interior of its closure, which is also equivalent to its algebraic interior : $\operatorname{Int} C=\operatorname{Int} \left(\operatorname{Cl} C\right)=C^{i}$. This intricate set of equalities highlights the deep connections between topological notions like interior and closure, and the algebraic properties inherent in convexity.

Convex hulls and Minkowski sums

These operations reveal how convex sets interact and combine, forming more complex structures while retaining their fundamental nature.

Convex hulls

Main article: convex hull

Every single subset A of a vector space, no matter how convoluted or disconnected, is inherently contained within a smallest possible convex set. This unique, minimal container is, as previously mentioned, termed the convex hull of A. It is rigorously defined as the intersection of all possible convex sets that encompass A. The operator that maps any set S to its convex hull, often denoted Conv(), exhibits the characteristic properties of what is known in mathematics as a closure operator :

Extensive: The original set S is always a subset of its convex hull, i.e., S $\subseteq$ Conv(S). You can’t make a set smaller by taking its convex hull.
Non-decreasing : If one set S is a subset of another set T, then the convex hull of S will also be a subset of the convex hull of T. Formally, S $\subseteq$ T implies that Conv(S) $\subseteq$ Conv(T). Larger sets produce larger (or equal) convex hulls.
Idempotent : Taking the convex hull of a convex hull yields no further change; it’s already convex. Thus, Conv(Conv(S)) = Conv(S). Once it’s convex, it stays convex under this operation.

These properties are not just academic curiosities; they are essential for the set of all convex sets to form a mathematical lattice . In this lattice structure, the “join” operation (analogous to a union) is precisely the convex hull of the union of two convex sets:

$${\displaystyle \operatorname {Conv} (S)\vee \operatorname {Conv} (T)=\operatorname {Conv} (S\cup T)=\operatorname {Conv} {\bigl (}\operatorname {Conv} (S)\cup \operatorname {Conv} (T){\bigr )}.}$$

Given that the intersection of any collection of convex sets is itself convex, it logically follows that the convex subsets of a (real or complex) vector space form a complete lattice . This implies a highly ordered and structured relationship between all possible convex sets within that space.

Minkowski addition

Main article: Minkowski addition

Minkowski addition is a fundamental operation that allows for the “addition” of geometric shapes, rather than just points. It’s a powerful tool for describing how shapes combine and interact.

(Insert image here, as it was in the original) Minkowski addition of sets. The sum of the squares $Q_1 = [0,1]^2$ and $Q_2 = [1,2]^2$ is the square $Q_1+Q_2 = [1,3]^2$.

In a real vector-space, the Minkowski sum of two non-empty sets, $S_1$ and $S_2$, is defined by adding every possible vector from $S_1$ to every possible vector from $S_2$, element-wise. The resulting set, $S_1 + S_2$, is given by:

$${\displaystyle S_{1}+S_{2}={x_{1}+x_{2}:x_{1}\in S_{1},x_{2}\in S_{2}}.}$$

More generally, for any finite family of non-empty sets $S_n$, their Minkowski sum is defined as the set formed by the element-wise addition of vectors from each set in the family:

$${\displaystyle \sum {n}S{n}=\left{\sum {n}x{n}:x_{n}\in S_{n}\right}.}$$

For the operation of Minkowski addition, the zero set—the set ${0}$ containing only the zero vector $0$—holds a special importance . It acts as the identity element: for any non-empty subset $S$ of a vector space, adding the zero set leaves $S$ unchanged: $S+{0}=S$. In algebraic terminology, ${0}$ serves as the identity element for Minkowski addition when applied to the collection of non-empty sets. (A brief, yet crucial, aside: the empty set behaves quite differently; for any subset $S$, $S+\emptyset=\emptyset$. It annihilates everything, which is a property worth noting if you’re into existential voids.)

Convex hulls of Minkowski sums

A particularly elegant result demonstrates how Minkowski addition interacts with the operation of taking convex hulls. These two operations, often complex on their own, behave remarkably well when combined. The following proposition highlights this harmonious relationship:

Let $S_1$ and $S_2$ be any subsets of a real vector-space. The convex hull of their Minkowski sum is precisely equal to the Minkowski sum of their individual convex hulls:

$${\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).}$$

This isn’t just a convenient formula; it’s a deep structural truth. This result extends even more generally to any finite collection of non-empty sets:

$${\displaystyle {\text{Conv}}\left(\sum {n}S{n}\right)=\sum {n}{\text{Conv}}\left(S{n}\right).}$$

In precise mathematical terminology, this means that the operations of Minkowski summation and of forming convex hulls are commuting operations. The order in which you apply them doesn’t change the final result, which, in mathematics, is often a sign of underlying elegance and simplicity.

Minkowski sums of convex sets

When the sets involved in Minkowski addition are themselves convex, further properties emerge regarding their topological characteristics. The Minkowski sum of two compact convex sets is invariably compact. Furthermore, the sum of a compact convex set and a closed convex set will always result in a closed set. These properties assure a certain stability under this operation.

However, the closure property of differences of convex sets is more nuanced. The following celebrated theorem, established by Dieudonné in 1966, provides a sufficient condition for the difference of two closed convex subsets to remain closed. To state this, one must first understand the concept of a recession cone of a non-empty convex subset $S$, which is defined as:

$${\displaystyle \operatorname {rec} S=\left{x\in X,:,x+S\subseteq S\right},}$$

where this set $\operatorname{rec} S$ is itself a convex cone that always contains $0 \in X$ and satisfies the property $S+\operatorname{rec} S=S$. Essentially, the recession cone captures the “directions” in which the set extends infinitely without changing its shape. It’s the set of vectors you can add to any point in $S$ and still remain within $S$. Note that if $S$ is closed and convex, then its recession cone $\operatorname{rec} S$ is also closed, and for any $s_0 \in S$, it can be expressed as:

$${\displaystyle \operatorname {rec} S=\bigcap {t>0}t(S-s{0}).}$$

Theorem (Dieudonné). Let A and B be non-empty, closed, and convex subsets of a locally convex topological vector space such that the intersection of their recession cones, $\operatorname{rec} A \cap \operatorname{rec} B$, forms a linear subspace. If either A or B is locally compact (a weaker condition than compact), then their difference, A − B, is guaranteed to be closed. This theorem is a sophisticated result, vital for understanding the behavior of convex sets in more complex analytical settings.

Generalizations and extensions for convexity

The fundamental notion of convexity, while rigorously defined for Euclidean spaces, can be fruitfully generalized. This typically involves modifying the core definition in various aspects, leading to a family of related concepts often grouped under the umbrella term “generalized convexity.” The resulting mathematical objects, though not strictly convex by the original definition, nonetheless retain certain analogous and useful properties of their more traditional counterparts. It’s an exploration of how far one can stretch a good idea before it breaks.

Star-convex (star-shaped) sets

Main article: Star domain

A common and intuitive relaxation of the convexity criterion leads to the concept of a star-convex, or star-shaped, set. Let C be a set in a real or complex vector space. C is deemed star convex if there exists at least one point $x_0$ within C such that the entire line segment drawn from $x_0$ to any other point $y$ in C is completely contained within C. Think of it as having a “central viewpoint” from which all other points are visible without leaving the set.

From this definition, it immediately follows that any non-empty convex set is, by default, also star-convex. After all, if any two points can be connected, then certainly a specific central point can connect to all others. However, the converse is emphatically not true: a star-convex set is not necessarily convex. A simple example is a star shape itself, or a ‘T’ shape; you can pick a central point that connects to all others, but you might not be able to connect two points on opposite “arms” of the star without leaving the shape. It’s a weaker, more permissive form of connectivity.

Orthogonal convexity

Main article: Orthogonal convex hull

Another intriguing example of generalized convexity is orthogonal convexity. This concept arises particularly in computational geometry and image processing, where axis-aligned structures are prevalent.

A set S within Euclidean space is designated as orthogonally convex or ortho-convex if any line segment that is parallel to any of the coordinate axes and connects two points within S, lies entirely within S. It’s convexity, but only along specific, cardinal directions. This property is more constrained than general convexity but is highly practical in certain applications. It is relatively straightforward to demonstrate that the intersection of any collection of orthoconvex sets will also be orthoconvex. Furthermore, several other properties analogous to those of standard convex sets hold true for orthoconvex sets, making them a useful class of objects for specialized analysis.

Non-Euclidean geometry

The fundamental definition of a convex set, along with its associated concept of a convex hull , naturally extends beyond the familiar confines of Euclidean geometry. This generalization is achieved by adapting the notion of a “straight line” to the specific geometry in question. In these more abstract settings, a set is defined as a geodesically convex set if it contains the geodesics joining any two points within the set. A geodesic, in essence, is the shortest path between two points in a given space, which may not appear “straight” in a Euclidean sense but is the equivalent of a straight line within that particular geometry (e.g., a great circle on a sphere). This extension allows the powerful framework of convexity to be applied to a much broader range of mathematical spaces, from Riemannian manifolds to metric spaces.

Order topology

Convexity can be extended even further, applying to a totally ordered set X that is endowed with an order topology . This is a highly abstract generalization, demonstrating the versatility of the core idea.

Let Y be a subset of X ($Y \subseteq X$). The subspace Y is defined as a convex set if, for every pair of points $a, b$ in Y such that $a \leq b$, the entire interval $[a, b] = {x \in X \mid a \leq x \leq b}$ is completely contained within Y. In simpler terms, Y is convex if and only if for all $a, b$ in Y, the condition $a \leq b$ implies that $[a, b] \subseteq Y$. This means there are no “gaps” in the set relative to the ordering.

However, a crucial distinction must be made: a convex set within an order topology is not necessarily connected in the topological sense, which might run counter to intuition derived from Euclidean spaces. A clear counter-example is provided by the subspace ${1, 2, 3}$ within the set of integers $\mathbb{Z}$ (with its usual order topology). This set is convex according to the definition above (any interval $[a,b]$ within ${1,2,3}$ is itself contained within ${1,2,3}$). Yet, it is clearly not connected, as the points are discrete and separated. This highlights that while the definition of convexity is extended, some topological properties do not directly translate.

Convexity spaces

The very notion of convexity can be elevated to an even higher level of abstraction by selecting certain fundamental properties of convexity and formalizing them as axioms . This leads to the concept of a “convexity space.”

Given a set X, a convexity over X is formally defined as a collection $\mathcal{C}$ of subsets of X that rigorously satisfy the following three axioms:

The empty set ($\emptyset$) and the entire set X are both members of $\mathcal{C}$. These are the trivial, yet necessary, elements of any convexity structure.
The intersection of any collection of sets drawn from $\mathcal{C}$ must also be a member of $\mathcal{C}$. This is the powerful closure under intersection that makes convex sets so well-behaved.
The union of a chain (a totally ordered set with respect to the inclusion relation ) of elements belonging to $\mathcal{C}$ must also be in $\mathcal{C}$. This ensures that convexity is preserved under certain types of “growth” or accumulation.

The elements of $\mathcal{C}$ are, by definition, called convex sets, and the pair $(X, \mathcal{C})$ is then referred to as a convexity space. For the ordinary, Euclidean definition of convexity, the first two axioms hold naturally, and the third one is trivially satisfied. This axiomatic approach provides a flexible framework for studying “convexity-like” structures in diverse mathematical contexts. For an alternative definition of abstract convexity, one that is perhaps more directly suited to the combinatorial nature of discrete geometry , one might investigate the convex geometries that are intrinsically associated with antimatroids .

Convex spaces

Main article: Convex space

Finally, convexity can be generalized as an abstract algebraic structure, leading to the definition of a convex space. In essence, a space is considered convex if it is possible, by definition, to form convex combinations of its points. This focuses on the operative aspect of convexity, providing a high-level algebraic perspective that underpins many of the geometric and topological manifestations.