Convexity is a property of shapes or functions where a line segment connecting any two points on the shape or curve lies entirely above or on the shape itself. In finance, convexity refers to the relationship between bond prices and interest rates, indicating how the price of a bond changes as interest rates fluctuate. In mathematics, convexity is important in optimization problems, as convex functions have unique minima, making them easier to analyze. Convex sets, which are collections of points where any line segment between two points remains within the set, are fundamental in various fields, including economics and geometry.