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.

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.

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.

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.

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.

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 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).

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 Saskatoon Experiment refers to a series of studies conducted in the 1970s and 1980s in Saskatoon, Saskatchewan, Canada, that focused on the effects of various nutritional interventions on mental health and behavior. These studies primarily investigated the role of diet in the management of conditions such as Attention Deficit Hyperactivity Disorder (ADHD) and other behavioral issues in children.

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.

Archeops is a dual-type Rock/Flying Pokémon introduced in Generation V of the Pokémon series. It is known as the "Archeops" Pokémon and is classified as the Fossil Pokémon. Evolving from Archen when it is revived from the Plume Fossil, Archeops is characterized by its bird-like appearance, featuring a crest on its head and vibrant plumage.

The Arcminute Cosmology Bolometer Array Receiver (ACBAR) is an astronomical instrument designed to measure the cosmic microwave background (CMB) radiation with high sensitivity and angular resolution. ACBAR primarily focuses on understanding the early universe and fundamental cosmological parameters, providing insights into the formation and evolution of cosmic structures.

The Arcminute Microkelvin Imager (AMI) is an innovative radio telescope designed to study the cosmic microwave background (CMB) and to investigate the large-scale structure of the universe. It operates in the microwave frequency range and is specifically aimed at measuring faint astronomical signals with high angular resolution and sensitivity.

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 **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.

AMiBA stands for the Arrayed Micro-Bolometer Array. It is an astronomical instrument designed for observing the cosmos, particularly in millimeter and submillimeter wavelengths. The primary goal of AMiBA is to study cosmic phenomena such as galaxy formation and evolution by mapping the cosmic microwave background (CMB) radiation and other astronomical objects. One of AMiBA's notable features is its use of an array of detector elements, which allows for high-resolution imaging and large-field observations.