Convex Body

Alright, if you insist on dragging me into this... Here's your expanded insight into the utterly captivating world of non-empty convex sets in Euclidean space. Don't expect fireworks; it's mathematics, not a circus.

Non-empty Convex Body in Euclidean Space

One might observe, with a minimum of effort, that a dodecahedron stands as a quintessential example of a convex body. It's a regular polyhedron composed of twelve regular pentagonal faces, and its inherent convexity is, frankly, undeniable to anyone with functional vision. Every line segment connecting any two points within its bounds remains entirely contained within the dodecahedron itself – a rather elegant, if somewhat predictable, geometric property.

In the realm of mathematics, a "convex body" in n- dimensional Euclidean space, denoted as R^n, is not just any old shape. It is, more precisely, a compact convex set that possesses a non-empty interior. This definition, while seemingly straightforward, carries a certain weight. The n- dimensional Euclidean space R^n itself is a fundamental construct, representing a space where points are n-tuples of real numbers, equipped with the standard Euclidean distance. Think of it as the familiar 2D plane or 3D space, simply generalized to an arbitrary number of dimensions.

The requirement for a set to be "compact" means it must be both closed and bounded. A closed set includes all its limit points, meaning it contains its "boundary." A bounded set can be entirely contained within some sufficiently large Euclidean ball – it doesn't stretch infinitely in any direction. These conditions are not arbitrary; they ensure the set behaves "nicely" from an analytical perspective, preventing pathologies that might arise from unboundedness or openness. For instance, a convex cone extending infinitely would be a convex set but not a convex body due to its lack of boundedness.

Furthermore, the defining characteristic of a "convex set" is that for any two points chosen within the set, the entire straight line segment connecting those two points must also lie entirely within the set. This is a remarkably intuitive geometric property; a crescent moon shape, for example, is not convex because one could easily draw a line segment between two points on its "horns" that passes outside the moon's body. A solid sphere, or indeed, our dodecahedron, perfectly satisfies this.

Now, the caveat: the stipulation of a non-empty interior. This particular clause ensures that a convex body is "full-dimensional" within R^n. It implies that the set contains an open Euclidean ball of some positive radius. A line segment in R^2, for example, is a compact convex set, but its interior in R^2 is empty – it's a 1-dimensional object embedded in a 2-dimensional space. Such an object would not be considered a convex body under this stricter definition. Some authors, in their infinite wisdom and quest for generality, choose not to impose the non-empty interior requirement, content with merely a non-empty set. This leads to slightly broader classes of objects but, predictably, complicates some theorems where the "thickness" of the body is implicitly assumed. For the purpose of most practical applications in convex geometry and optimization, the non-empty interior is a standard and useful assumption.

Symmetric Convex Bodies

A convex body K earns the descriptor "symmetric" if it exhibits central symmetry with respect to the origin. To be less opaque about it, this means that if a point x is an inhabitant of K, then its antipode, -x, must also reside within K. One might imagine rotating the set 180 degrees around the origin, and it would appear indistinguishable from its original configuration. This property is more than just an aesthetic quirk; it's a fundamental characteristic that establishes a profound and rather elegant one-to-one correspondence between these symmetric convex bodies and the unit balls of norms on R^n.

Let's unpack that, because it's genuinely useful. A norm on R^n is a function that assigns a "length" or "magnitude" to each vector, satisfying three key properties: it's positive definite (only the zero vector has zero length), it's homogeneous (scaling a vector scales its length by the same factor), and it obeys the triangle inequality (the length of the sum of two vectors is no more than the sum of their lengths). The unit ball of a norm ||.|| is simply the set of all vectors x such that ||x|| <= 1. It is a fundamental result that any such unit ball is always a compact, convex set that is centrally symmetric about the origin, thus qualifying it as a symmetric convex body.

Conversely, and this is where the "correspondence" truly shines, any symmetric convex body K (provided it contains the origin in its interior) can be used to define a norm. This norm, often called the Minkowski functional, is given by ||x||_K = inf{λ > 0 : x ∈ λK}. In essence, ||x||_K is the smallest factor λ by which you need to scale K so that x lies on its boundary. This reciprocal relationship is a cornerstone of functional analysis and convex geometry, providing a powerful bridge between geometric shapes and algebraic structures.

Some commonly known examples of these geometric entities, which you've likely encountered without realizing their profound implications, include:

The Euclidean ball: This is the familiar sphere in 3D, or a disk in 2D, or its higher-dimensional analogue. It is the unit ball for the standard Euclidean norm (the L2 norm), where ||x||_2 = sqrt(sum(x_i^2)). Its perfect symmetry and smoothness make it an ideal starting point for many geometric investigations.
The hypercube: In R^n, this is the generalization of a square (2D) or a cube (3D). It consists of all points x = (x1, ..., xn) where |xi| <= 1 for all i. This shape is the unit ball for the L∞ norm (or maximum norm), defined as ||x||_∞ = max(|x1|, ..., |xn|). It's a polyhedral body, characterized by flat faces and sharp corners, representing a distinct geometric flavor compared to the smooth Euclidean ball.
The cross-polytope: This is the dual of the hypercube. In 2D, it's a square rotated 45 degrees (a diamond); in 3D, it's an octahedron. It consists of all points x = (x1, ..., xn) such that sum(|xi|) <= 1. This is the unit ball for the L1 norm (or Manhattan norm), ||x||_1 = sum(|xi|). Just like the hypercube, it's a polyhedral symmetric convex body, but with a different set of vertices and faces, demonstrating the rich variety within this class of geometric objects.

Metric Space Structure

Let's denote the grand collection of all convex bodies in R^n as K^n. One might be tempted to think of these simply as individual shapes, but it's more productive to consider them as points within a larger, more abstract space. And, as it turns out, K^n is not just any collection; it's a complete metric space when equipped with a particular metric, often referred to as the Hausdorff distance. This metric, d(K,L), quantifies the "distance" between two convex bodies K and L. It's defined as the infimum (the greatest lower bound) of all ε ≥ 0 such that K is contained within L "fattened" by ε, and L is contained within K "fattened" by ε.

More formally, d(K,L) := inf{ε ≥ 0: K ⊂ L+B^n(ε), L ⊂ K+B^n(ε)}. Here, B^n(ε) represents an open Euclidean ball of radius ε centered at the origin. The notation L+B^n(ε) signifies the Minkowski sum of L and B^n(ε). Geometrically, this operation "fattens" the set L by adding to each point in L every point in B^n(ε). Essentially, it's like surrounding L with a uniform layer of thickness ε. So, the Hausdorff distance d(K,L) tells us the minimum ε required for K to be "almost" contained in L, and L to be "almost" contained in K. If two convex bodies are geometrically very similar, this ε will be small, indicating they are "close" in this metric space. This distance provides a rigorous way to measure the resemblance between shapes, which is, admittedly, rather clever.

The fact that K^n forms a complete metric space under this metric is not merely a mathematical curiosity. A complete metric space is one where every Cauchy sequence converges to a limit point that is also within the space. This property is immensely valuable in analysis, as it guarantees the existence of solutions to various problems. For instance, if you have a sequence of convex bodies that are getting progressively "closer" to each other in a specific way, you are assured that they are converging to a well-defined convex body within K^n, rather than to some non-convex entity or a set that "escapes" the space.

Adding another layer of analytical convenience, the Blaschke Selection Theorem states that any d-bounded sequence in K^n possesses a convergent subsequence. To translate this from the esoteric, a d-bounded sequence means that all the convex bodies in the sequence can be contained within some fixed, larger Euclidean ball (they don't fly off to infinity), and they all contain some fixed, smaller Euclidean ball (they don't shrink to a point or degenerate into a lower-dimensional object). The theorem then guarantees that even if the original sequence doesn't converge, you can always pick a sub-sequence from it that does converge to a convex body. This is a powerful result, often invoked in proofs concerning existence or approximation in convex geometry, akin to the Bolzano–Weierstrass theorem for sequences of real numbers, ensuring that within a bounded and closed set, you can always find a convergent path. It means the space of convex bodies, when appropriately constrained, behaves rather compactly itself. [1]

Polar Body

If you're looking for a way to transform one convex body into another, often with surprising insights into its original structure, then the concept of a polar body might pique your interest – or at least marginally hold it. Given a bounded convex body K that crucially contains the origin O within its interior, its polar body, denoted K^*, is defined as the set of all vectors u such that their dot product (or inner product) with any vector v from K does not exceed 1. Mathematically, this is expressed as: K^* = {u : ⟨u,v⟩ ≤ 1, ∀v ∈ K}.

The conditions for K are not arbitrary limitations; they are essential. K being bounded ensures K^* is also bounded, and K containing the origin in its interior guarantees that K^* is also a convex body with a non-empty interior, preventing it from collapsing into a degenerate object. The dot product ⟨u,v⟩ measures the projection of one vector onto another, scaled by their magnitudes. Geometrically, the inequality ⟨u,v⟩ ≤ 1 defines a half-space whose boundary is a hyperplane perpendicular to u and passing at a distance 1/||u|| from the origin. The polar body K^* is, therefore, the intersection of an infinite number of such half-spaces, one for each v ∈ K. This construction inherently results in a convex set.

The polar body operation possesses several rather "nice" properties, which elevate it beyond a mere definition to a fundamental tool in convex analysis and duality theory:

Involutory Property: (K^*)^* = K. This means that if you take the polar body of K, and then take the polar body of that result, you get back to your original set K. This reflexivity is a hallmark of a robust duality relation, signifying that the transformation is, in a sense, its own inverse. It's like looking into a mirror and then looking into a mirror of that reflection; you eventually see yourself again.
Boundedness: K^* is bounded. This is a direct consequence of K containing the origin in its interior. If K contains a small ball around the origin, then K^* must be bounded, as u cannot be arbitrarily large if ⟨u,v⟩ ≤ 1 for all v in that small ball.
Order-Reversing Inclusion: If K1 ⊂ K2 then K2^* ⊂ K1^*. This is an intriguing property: if one convex body is contained within another, their polar bodies exhibit the reverse inclusion. Intuitively, if K2 is larger than K1, then the vectors v in K2 can be "longer" or extend further from the origin. To satisfy ⟨u,v⟩ ≤ 1 for these potentially larger vs, the vectors u in K2^* must necessarily be "shorter" or more constrained than those in K1^*. This inversion of inclusion is a powerful characteristic of polar duality.

In essence, the polar body provides a sophisticated type of duality relation, transforming a convex body into another that encodes information about its "support" or "width" in different directions. This concept is indispensable in areas such as optimization, geometry of numbers, and functional analysis, offering an alternative perspective on the geometric properties of sets.

Convex Body

Non-empty Convex Body in Euclidean Space

Symmetric Convex Bodies

Metric Space Structure

Polar Body

See also