Convex geometry

Convex geometry is a branch of mathematics that studies convex sets and their properties in various dimensions. A set is defined as convex if, for any two points within the set, the line segment connecting those two points lies entirely within the set. This simplicity in definition leads to rich geometric and combinatorial properties.

Asymptotic geometric analysis is a branch of mathematics that combines techniques from geometry, functional analysis, and asymptotic analysis to study the geometric properties of spaces, particularly in the context of high-dimensional analysis. It often focuses on how geometric structures behave as dimensions grow large or as certain parameters tend to infinity.

A **convex hull** is a fundamental concept in computational geometry. It can be defined as the smallest convex set that contains a given set of points in a Euclidean space. To visualize it, imagine stretching a rubber band around a set of points on a plane; when the band is released, it will form a shape that tightly encloses all the points. This shape is the convex hull of that set of points.

Geometric transversal theory is a branch of mathematics and combinatorial geometry that deals with the study of transversals in geometric settings, particularly in relation to point sets and geometric objects like lines, segments, or more general shapes. The study often involves finding intersections, arrangements, and coverings that satisfy certain combinatorial conditions.

Oriented matroids are a combinatorial structure that generalizes the concept of linear independence in vector spaces to a broader context. They arise in the study of combinatorial geometry and optimization and have applications in various fields such as discrete geometry, algebraic geometry, and matroid theory. ### Definition: An oriented matroid can be thought of as a matroid (a structure that generalizes the notion of linear independence) equipped with an additional orientation that indicates the “direction” of independence among its elements.

Polyhedra are three-dimensional geometric figures with flat polygonal faces, straight edges, and vertices (corners). The word "polyhedron" comes from the Greek words "poly," meaning many, and "hedron," meaning face. Each face of a polyhedron is a polygon, a two-dimensional shape with straight sides.

Polytopes are geometric objects that exist in any number of dimensions and have flat sides (called faces). In a more formal mathematical sense, a polytope is defined as the generalized version of polygons (2D) and polyhedra (3D). Here are some key points about polytopes: 1. **Dimensions**: - A **polygon** is a 2-dimensional polytope (e.g., triangles, squares).

Convex geometry is a branch of mathematics that studies convex sets and their properties. It encompasses a variety of theorems that address the structure, behavior, and relationships of convex sets and functions.

An **antimatroid** is a combinatorial structure that generalizes certain properties of matroids. It is defined by a collection of sets that satisfy specific axioms.

A **B-convex space** is a concept from functional analysis and convex analysis that generalizes the idea of convexity in mathematical spaces. In a B-convex space, the traditional notion of convex combinations is extended to allow for certain types of structured combinations of points.

Betavexity is not a universally recognized term in finance, economics, or other common fields. It may relate to a specific concept, product, or a term used in niche circles or emerging trends that have arisen after my last knowledge update in October 2021.

Bond convexity is a measure of the curvature in the relationship between bond prices and bond yields. It builds upon the concept of duration, which measures the sensitivity of a bond's price to changes in interest rates. While duration gives a linear approximation of price changes for small changes in yield, convexity provides a more accurate measure by accounting for the curvature in this relationship.

The Busemann–Petty problem is a classic question in the field of convex geometry. It asks whether, in Euclidean space, the volume of a convex body can be deduced solely from the volumes of its orthogonal projections onto a hyperplane. More specifically, if two convex bodies have the same volume for all orthogonal projections, do they necessarily have to be congruent (that is, identical up to rigid motion)?

A **conical combination** is a mathematical concept primarily used in linear algebra and geometry. It refers to a specific type of linear combination of points (or vectors) that satisfies certain constraints, particularly in relation to convexity.

A convex polytope is a geometric object that exists in a finite-dimensional space (typically in Euclidean space). It is defined as the convex hull of a finite set of points, which means it is the smallest convex set that contains all those points.

Convex analysis is a branch of mathematical analysis that studies the properties of convex sets and convex functions. It is an important area in various fields, including optimization, economics, and functional analysis. The main focus of convex analysis is understanding how convex structures facilitate various mathematical and practical problems.

A convex body is a specific type of geometric figure in Euclidean space that possesses certain characteristics. Formally, a convex body can be defined as follows: 1. **Compactness**: A convex body is a compact set, meaning it is closed and bounded.

A **convex combination** is a specific type of linear combination of points (or vectors) where the coefficients are constrained to be non-negative and sum to one.

A convex curve is a type of curve in mathematics that has the property that any line segment drawn between two points on the curve lies entirely within or on the curve itself. This means that if you take any two points on the curve and connect them with a straight line, the entire line segment will not cross outside of the curve. Key properties of convex curves include: 1. **Non-Concavity**: A convex curve does not curve inward at any point. Instead, it always bows outward.

A **convex metric space** is a concept from the field of metric geometry, which generalizes the idea of convexity in Euclidean spaces to more abstract metric spaces. In a convex metric space, the notion of "straight lines" between points is defined in terms of the metric, allowing one to discuss the convexity of sets and the existence of curves connecting points.

A convex polygon is a type of polygon in which all its interior angles are less than 180 degrees. This characteristic means that any line segment drawn between two points within the polygon will lie entirely inside the polygon. Additionally, for a convex polygon, for any two points within the polygon, the straight line connecting them does not exit the polygon at any point. Key properties of convex polygons include: 1. **Interior Angles**: Each interior angle is less than 180 degrees.

A **convex polytope** is a mathematical object that generalizes the concept of polygons and polyhedra to higher dimensions. More formally, a convex polytope can be defined in several ways, including: 1. **Geometrically:** A convex polytope is a bounded subset of Euclidean space that is convex, meaning that for any two points within the polytope, the line segment connecting them is also contained within the polytope.

In mathematics, particularly in the field of convex analysis, a **convex set** is defined as a subset \( C \) of a vector space such that, for any two points \( x \) and \( y \) in \( C \), the line segment connecting \( x \) and \( y \) is also entirely contained within \( C \).

In finance, **convexity** refers to the curvature in the relationship between bond prices and bond yields. It is a measure of how the duration of a bond changes as interest rates change, and it helps investors understand how the price of a bond will react to interest rate fluctuations. Here are key points to understand convexity: 1. **Price-Yield Relationship:** The relationship between bond prices and yields is not linear; thus, the price does not change at a constant rate as yields change.

In economics, convexity refers to the shape of a curve that represents a relationship between two variables, typically in the context of utility functions, production functions, or cost functions. The concept of convexity is crucial in understanding optimization problems, consumer behavior, and market dynamics. Here are some key points about convexity in economics: 1. **Utility Functions**: A utility function is said to be convex if it exhibits diminishing marginal utility.

A **Difference Bound Matrix (DBM)** is a data structure used primarily in the analysis of timed automata, which are models used in formal verification and automatic synthesis of systems with timing constraints. The DBM is particularly useful for representing relationships between time constraints in a compact way. ### Key Features of Difference Bound Matrices: 1. **Matrix Representation**: A DBM is typically represented as a matrix where each entry corresponds to the difference between two clocks (or variables).

In the context of convex analysis and optimization, the concepts of the dual cone and polar cone are important tools used to study properties of convex sets and relationships between them.

Dykstra's projection algorithm is an iterative method used in convex optimization for finding the projection of a point onto the intersection of convex sets. It is particularly useful because it efficiently handles scenarios where the intersection is defined by multiple convex sets, and it can be used in applications such as signal processing, image reconstruction, and statistics.

The Equichordal Point Problem is a problem in the field of geometry and optimization that involves finding a point in a given arrangement of chords in a circle such that the sum of the distances from that point to each of the chords is minimized.

"Exposed Point" can refer to different concepts depending on the context, such as in mathematics, geography, or other fields. However, this term isn't universally defined as a standard term across disciplines. Here are some possible interpretations: 1. **Mathematics/Geometry**: In geometrical contexts, an exposed point can refer to a point on a polyhedron or surface that is not obscured by other parts of the shape.

An "extreme point" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Mathematics (Geometry)**: In the context of convex sets, an extreme point of a convex set is a point in that set that cannot be expressed as a convex combination of other points in the set. For example, in a polygon, the vertices are extreme points because they cannot be represented as a combination of other points in the polygon.

In geometry, a "face" is a flat surface that forms part of the boundary of a solid object. Faces are the two-dimensional shapes that make up the surfaces of three-dimensional figures, such as polyhedra. Each face is typically a polygon, and the arrangement of these faces defines the overall shape of the solid. For example: - A cube has six square faces. - A triangular prism has two triangular faces and three rectangular faces. - A tetrahedron has four triangular faces.

The Gilbert–Johnson–Keerthi (GJK) distance algorithm is a computational geometry algorithm used for determining the distance between convex shapes in space, particularly in robotics and computer graphics. It is widely utilized for collision detection, where understanding the proximity of objects is essential. ### Key Features of the GJK Algorithm: 1. **Convex Shapes**: The GJK algorithm is specifically designed for convex shapes.

The John ellipsoid is a specific type of ellipsoid that is used in the context of convex analysis and optimization. It is associated with the John’s theorem, which deals with the geometry of convex bodies. More formally, the John ellipsoid of a convex body \( K \) in \( \mathbb{R}^n \) is the unique ellipsoid of maximal volume that can be inscribed in \( K \).

The Klee–Minty cube is a specific example of a convex polytope that is often used in the context of linear programming and optimization problems. It is particularly known for its role in demonstrating the limitations of certain types of algorithms, especially the simplex method. The Klee–Minty cube is an example of a "non-simple" polytope, which means that it has many facets but can be difficult for simplex methods to optimize in a straightforward manner.

In geometry, a lens is a shape formed by the intersection of two circular arcs. Specifically, it is the region bounded by two circles that overlap. The area enclosed by these arcs resembles the shape of a lens, which is the reason for its name. There are two main types of lenses: 1. **Convex Lens**: This occurs when both arcs are part of circles that are convex towards each other. The resulting lens shape bulges outward.

Convexity is a rich and multifaceted area of study in mathematics and related fields. Here’s a list of key topics related to convexity: 1. **Basic Definitions:** - Convex sets - Convex functions - Strictly convex functions 2.

The Mahler volume is a concept from the field of convex geometry and number theory. Specifically, it refers to a particular measure associated with a multi-dimensional geometric shape called a convex body. The Mahler volume \( M(K) \) of a convex body \( K \) in \( n \)-dimensional space is defined as the product of the volume of the convex body and the volume of its polar body.

Minkowski Portal Refinement (MPR) is a computational method used in materials science and crystallography for the analysis of crystalline structures. It combines geometric and optimization principles to explore the configuration space of possible atomic arrangements within a given material, particularly for complex or disordered systems. The method is named after Hermann Minkowski, who contributed to the field of geometry and mathematical formulations that are relevant in crystallography.

Mixed volume is a concept in the field of algebraic geometry and convex geometry, specifically in the study of polytopes and their measures. It generalizes the notion of volume to sets that may not be convex and provides a way to measure the "size" of a collection of convex bodies in a vector space.

In economics, non-convexity refers to a situation where the set of feasible outcomes or preferences does not maintain the property of convexity. To understand this concept better, it's essential to grasp what convexity means in this context. **Convexity**: A set is convex if, for any two points within that set, the entire line segment connecting them also lies within the set.

A projection body is a concept from convex geometry. It refers to a geometric object that is derived from a given convex body by considering its orthogonal projections onto various subspaces.

Projections onto convex sets is a mathematical concept often used in optimization, functional analysis, and convex geometry. The idea centers around finding a point in a convex set that is closest to a given point outside that set.

In the context of model checking, a "Region" typically refers to a specific approach or technique used for identifying and analyzing subsets of the state space of a system being modeled. Model checking itself is an automated technique used to verify that a model of a system meets certain specifications, typically expressed in temporal logic. The concept of regions is most commonly associated with the analysis of hybrid systems and real-time systems.

Rotating calipers is a computational geometry technique used primarily for solving problems related to convex shapes, particularly convex polygons. The method helps in efficiently calculating various geometric properties, such as distances, diameters, and optimizing certain geometric operations. ### Key Concepts of Rotating Calipers: 1. **Convex Hull**: The method is typically applied to the convex hull of a set of points in the plane, which is the smallest convex polygon that can enclose all the points.

The Shapley–Folkman lemma is a result in the field of convex analysis and mathematical economics. It is named after Lloyd S. Shapley and Stephen Folkman, who contributed to its development. The lemma provides insights into how the aggregation of small perturbations of a set can approximate a convex set.

Shephard's problem refers to a question in the field of convex geometry, specifically related to the properties of convex bodies and their projections. Named after the mathematician G. A. Shephard, the problem explores the relationship between the structure of a convex body in higher-dimensional spaces and the geometric properties of its projections in lower-dimensional spaces. In precise terms, Shephard's problem can be stated about the expected volume or surface area of projections of convex bodies onto lower-dimensional subspaces.

The term "support function" can refer to different concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Business Context**: In a business or organizational setting, support functions are departments or activities that assist the core operations of the business. Examples include human resources, IT support, customer service, and finance. These functions do not directly contribute to the production of goods or services but provide essential services that enable the core functions to operate smoothly.

A **supporting hyperplane** is a concept from convex analysis and geometry, particularly in the context of convex sets and optimization. It relates to how we can visualize and understand the boundaries of convex sets in multidimensional spaces. Formally, a hyperplane can be defined as a flat, affine subspace of one dimension less than the dimension of the surrounding space. For example, in a 3-dimensional space, a hyperplane is a 2-dimensional plane.

A **symmetric cone** is a special type of geometric cone that arises in the context of convex analysis and algebraic geometry. More formally, a symmetric cone can be defined as a proper, closed, convex cone in a finite-dimensional real vector space that has a certain invariance property under linear transformations. Symmetric cones are characterized by the following properties: 1. **Self-Duality**: A symmetric cone is self-dual, which means that the cone is equal to its dual cone.

In mathematical optimization and differential geometry, the **tangent cone** at a point \( x_0 \) of a set \( C \) is a concept that describes the directions in which one can move from that point while remaining within the set. It is particularly useful in the study of convex analysis, nonsmooth analysis, and variational analysis.