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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
"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.
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.