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Convex geometry is the branch of mathematics that studies convex sets—subsets of a vector space that contain, along with any two of their points, the entire line segment joining them. This single, deceptively simple condition gives rise to a remarkably rich theory that connects linear algebra, analysis, combinatorics, and optimization. The field asks foundational questions about the shape, size, and structure of these sets, and its answers have become indispensable tools across mathematics and its applications.
A convex set is defined by a closure property: if points \(x\) and \(y\) belong to the set, then every point of the form \((1-t)x + ty\) for \(0 \le t \le 1\) also belongs to it. This definition makes sense in any real vector space, but the most studied settings are finite-dimensional Euclidean space \(\mathbb{R}^n\), where geometric intuition is strongest, and infinite-dimensional Banach spaces, where analytic techniques are required.
The central questions of convex geometry revolve around several recurring themes:
The study of convexity has ancient roots. The Greeks understood convex polygons and the convexity of circles and spheres, and Archimedes computed volumes of convex solids. However, these were isolated results rather than a systematic theory. The modern field began to take shape in the late nineteenth and early twentieth centuries, when mathematicians started to treat convexity as a subject in its own right.
Hermann Minkowski is often regarded as the founder of the field. Around the turn of the twentieth century, he developed the theory of convex bodies—compact convex sets with nonempty interior—in Euclidean space. He introduced the support function, the Minkowski sum, and the mixed volumes that arise when one expands a convex body by adding a small multiple of another. His work was motivated partly by number theory (the geometry of numbers, which studies lattice points in convex sets) and partly by the foundations of geometry.
In the 1930s and 1940s, the subject expanded in two directions. On the one hand, the Polish school, led by Hugo Steinhaus and Stanisław Mazur, and the Soviet school, led by Leonid Kantorovich and others, developed the analytic and functional-analytic aspects of convexity, connecting it to the theory of normed spaces and to optimization. On the other hand, the work of Wilhelm Blaschke and his students in Germany brought a strong geometric and differential flavor, studying the curvature of convex surfaces and the stability of geometric inequalities.
A major conceptual shift occurred with the work of Werner Fenchel and Hans Rademacher in the 1930s, who systematically developed the theory of convex functions—functions whose epigraph (the set of points above the graph) is convex. This connected convex geometry to analysis in a fundamental way, leading to the theory of subgradients, Legendre transforms, and duality in optimization. The 1950s and 1960s saw the consolidation of these ideas in the work of Richard Bellman, Harold Kuhn, Albert Tucker, and others, who placed convexity at the center of mathematical programming.
The late twentieth century brought a new wave of results, particularly through the work of Alexander Dinghas, Erwin Lutwak, and others, who developed the \(L_p\) Brunn–Minkowski theory—a far-reaching generalization that replaces ordinary Minkowski addition with a family of operations parametrized by a real number \(p\). This theory has become a central organizing framework in the field, connecting convex geometry to information theory, probability, and even the geometry of Banach spaces.
While convex geometry is a unified subject, several distinct approaches have shaped its development, each with its own questions, methods, and emphases.
This approach treats convex sets as geometric objects in Euclidean space, emphasizing their shape, boundary structure, and combinatorial properties. Its methods are often elementary but subtle, relying on careful geometric reasoning, inequalities, and the theory of polytopes.
The study of polytopes—convex hulls of finitely many points—is a central part of this tradition. Polytopes have a rich combinatorial structure: their faces (vertices, edges, facets) form a partially ordered set that encodes much of their geometry. The Euler characteristic, which relates the numbers of faces of different dimensions, is a classical result that extends far beyond convexity. The classification of regular polytopes, the study of their symmetry groups, and the enumeration of polytopes with given properties are ongoing concerns.
A key achievement of this tradition is the theory of mixed volumes. Given convex bodies \(K1, \ldots, Km\) in \(\mathbb{R}^n\), the volume of the Minkowski sum \(t1 K1 + \cdots + tm Km\) is a homogeneous polynomial in the nonnegative variables \(t_i\). Its coefficients are the mixed volumes, which measure the "interaction" between the bodies. The Brunn–Minkowski inequality and its many consequences—such as the isoperimetric inequality, which states that among all bodies of a given volume, the ball minimizes surface area—are expressed naturally in this language.
This tradition also includes the study of geometric inequalities more broadly: the Brunn–Minkowski inequality, the isoperimetric inequality, the Sobolev inequalities, and their stability versions. A recurring theme is the identification of equality cases: when does a geometric inequality become an equality, and what does that say about the sets involved? These questions often lead to deep rigidity results.
This approach treats convexity through the lens of analysis, focusing on convex functions, duality, and the geometry of normed spaces. Its methods come from functional analysis, measure theory, and the calculus of variations.
The theory of convex functions is built on the observation that a function \(f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}\) is convex if and only if its epigraph is a convex set. This dictionary between functions and sets allows techniques from one domain to be imported into the other. The subgradient—a generalization of the derivative that exists even at points where a convex function is not differentiable—plays a central role. The Legendre–Fenchel transform, which maps a convex function to its dual, is the analytic counterpart of the support function and underlies the duality theory of optimization.
In infinite dimensions, convexity is intimately connected to the geometry of Banach spaces. A Banach space is said to be strictly convex if its unit ball has no nontrivial line segments on its boundary, and uniformly convex if the boundary curves uniformly. These properties have profound consequences for the existence and uniqueness of best approximations, the convergence of iterative methods, and the structure of the space's dual. The work of James, Lindenstrauss, and others in the mid-twentieth century established deep connections between the convex geometry of a Banach space and its linear structure.
This tradition also encompasses the theory of convex bodies in high-dimensional spaces, where the focus shifts from individual bodies to the typical behavior of bodies in many dimensions. The concentration of measure phenomenon—the observation that in high dimensions, most of the volume of a convex body lies near its boundary—has become a central theme, with applications to probability, statistics, and computer science.
This approach studies convex sets through their interactions with random objects: random lines, random points, random hyperplanes. Its methods come from integral geometry, geometric probability, and stochastic geometry.
The foundational result here is Crofton's formula, which expresses the length of a curve in terms of the expected number of intersections with random lines. For convex bodies, this generalizes to a family of formulas relating intrinsic volumes to averages over random flats. The kinematic formula of Blaschke and Santaló describes how the expected volume of the intersection of a moving convex body with a fixed one depends on the motion.
This tradition also includes the study of random polytopes—convex hulls of random points—and their expected properties. Questions about the expected number of vertices, the expected volume, and the fluctuations around these expectations have been studied extensively, with connections to the theory of random matrices and to computational geometry.
The stochastic tradition has important applications in stereology, where one must estimate properties of a three-dimensional body from measurements on random planar sections, and in the analysis of spatial point processes, where convexity appears in the study of Voronoi cells and Delaunay triangulations.
This approach treats convex sets as objects to be computed with: to be represented, manipulated, and optimized over. Its methods come from computational geometry, numerical analysis, and optimization theory.
The representation of convex sets is a central concern. Polytopes can be described either by their vertices (the \(\mathcal{V}\)-representation) or by their defining inequalities (the \(\mathcal{H}\)-representation), and converting between these descriptions is a fundamental algorithmic problem. For general convex sets, one often works with separation oracles—algorithms that, given a point, either confirm it lies in the set or produce a hyperplane separating it from the set. The ellipsoid method and interior-point methods for convex optimization are built on such representations.
The computational tradition has its own central questions: How efficiently can one determine whether a point lies in a convex set? How well can one approximate the volume of a convex body given only a separation oracle? The latter question was answered by Dyer, Frieze, and Kannan in the late 1980s, who showed that randomized algorithms can approximate the volume of a convex body in polynomial time, using techniques from Markov chain Monte Carlo. This result, and the subsequent development of the theory of sampling from log-concave distributions, has created deep connections between convex geometry and theoretical computer science.
These traditions are not isolated silos; they interact constantly and often fruitfully. The geometric and analytic traditions are linked by the dictionary between convex sets and convex functions, and by the fact that many geometric inequalities have analytic proofs and vice versa. The integral-geometric tradition provides tools—like Crofton's formula—that are used throughout the field, and its results often have purely geometric formulations. The computational tradition draws on all of the others: algorithms for convex optimization rely on the duality theory from the analytic tradition, while the analysis of random polytopes uses the integral-geometric tradition's tools.
A notable example of this interaction is the \(L_p\) Brunn–Minkowski theory, which emerged from the geometric tradition but has been developed using analytic, integral-geometric, and computational methods. This theory replaces the Minkowski sum with a family of operations that interpolate between the classical sum and the convex hull of the union, and it has led to new inequalities, new notions of curvature, and new connections to information theory.
Convex geometry today is a vibrant and expanding field, with several active research fronts. One major direction is the study of high-dimensional convex bodies, where the focus is on understanding the typical behavior of bodies as the dimension grows. This includes the theory of isotropic bodies, which are bodies whose mass is spread evenly in all directions, and the study of their covariance matrices. The Kannan–Lovász–Simonovits conjecture, which concerns the isoperimetric profile of isotropic bodies, remains open and is one of the most famous problems in the field, with connections to sampling, optimization, and the geometry of Banach spaces.
Another active direction is the development of the \(Lp\) Brunn–Minkowski theory and its relatives, including the recently developed \(Lp\) dual theory. These theories have produced new inequalities, such as the \(Lp\) Sobolev inequalities and the \(Lp\) affine isoperimetric inequalities, which have found applications in analysis and probability.
The theory of valuations—functions on convex bodies that are finitely additive under Minkowski addition—has seen a remarkable revival. Hadwiger's theorem, which classifies all continuous, rigid-motion-invariant valuations on convex bodies in \(\mathbb{R}^n\), is a classical result that has been extended in many directions. The modern theory of valuations connects convex geometry to algebraic geometry, representation theory, and even the theory of operads.
Finally, the computational aspects of convex geometry continue to develop, driven by applications in machine learning, statistics, and optimization. The problem of sampling from log-concave distributions, the analysis of convex relaxations for combinatorial problems, and the development of faster algorithms for convex optimization are all active areas where convex geometry plays a central role.
Convex geometry is thus a field unified by a single, simple definition but diversified by the many ways in which that definition can be explored. Its enduring appeal lies in the fact that convexity is both ubiquitous—appearing in optimization, probability, economics, and physics—and tractable, admitting a rich theory that is both deep and applicable. The field continues to grow, not by replacing old approaches with new ones, but by weaving them together into an ever more complete picture of the geometry of convex sets.