An interior point is a point that lies within the boundary of a set in a topological space, meaning there exists a neighborhood entirely contained within the set. This concept is fundamental in topology and optimization, where it is used to determine feasible regions and solutions within constraints.
Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.
Optimization is the process of making a system, design, or decision as effective or functional as possible by adjusting variables to find the best possible solution within given constraints. It is widely used across various fields such as mathematics, engineering, economics, and computer science to enhance performance and efficiency.
A convex body is a compact convex set with non-empty interior in a Euclidean space, meaning it is a shape where, for any two points within the shape, the line segment connecting them is entirely contained within the shape. Convex bodies are fundamental in geometry and optimization, serving as the building blocks for understanding more complex structures and problems in these fields.
A boundary point of a set in a topological space is a point where every neighborhood contains at least one point from the set and one point not in the set. This concept is crucial in understanding the structure and properties of topological spaces, as it helps delineate the 'edges' of a set within the space.