A shape is considered convex if, for any two points within the shape, the line segment connecting them lies entirely inside the shape. This means that a convex shape has no indentations or "dents." Common examples of convex shapes include circles, squares, and triangles. In contrast, a shape is non-convex if it has at least one indentation, causing some line segments between points to fall outside the shape. Understanding convexity is important in various fields, including geometry, computer graphics, and optimization, where it helps in analyzing shapes and solving problems efficiently.