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Closed convex sets are foundational to the study of convex optimization, a branch of mathematics that is critical to many areas, including economics, engineering, and machine learning.

An excellent starting point to understand closed convex sets is visualizing a filled geometric shape, such as a circle or a polygon, that does not contain any 'holes' or 'gaps'. Mathematically, a set is convex if, for any two points within the set, the line segment connecting these points is entirely contained within the set. A set is closed if it includes all its boundary points – in essence, if you can 'walk' on the edge of the shape without stepping outside of it.

In the context provided, a filled circle and an annulus are solid examples of closed convex sets. This is because every point inside these shapes, along with the points on the boundary, make up the set, and the line segment between any two points within these shapes does not escape their boundaries. Thus, they satisfy both conditions: being convex and being closed.

The concept of strict separation in convex optimization deals with distinguishing two sets using a 'hyperplane' without touching either set. Think of it as drawing a line between two non-overlapping shapes on a piece of paper, where the line is not allowed to touch either shape.

For strict separation to be possible, there should exist at least a small 'gap' or 'buffer zone' between the two sets. This allows a hyperplane to fit snugly in between without intersecting any point of either set. However, as illustrated in the example with the filled circle and the annulus, when one set fits entirely within the boundary of another without any gap, strict separation is not feasible because any line drawn will touch at least one of the sets, intruding upon the condition required for strict separation.

Disjoint convex sets are an interesting aspect of study in convex optimization. Two sets are disjoint if they do not share any elements – they have nothing in common, no points where they overlap.

Characteristics of Disjoint Convex Sets

Disjoint convex sets can be easily illustrated through simple shapes that do not intersect, such as two non-touching circles or rectangles. In our textbook example, a circle (A) and an annulus (B) act as representations of disjoint convex sets because there isn't a single point that both sets share. Despite this, the sets themselves are nicely rounded (convex), and because they include their boundaries, they are also closed.

Understanding disjoint convex sets is important in many practical optimization problems, where the solutions often involve separating resources or preferences that cannot coexist in the same space or condition.

Convexity plays a pivotal role in the field of optimization, as it simplifies the process of finding optimal solutions. In convex optimization, the goal is often to find the lowest or highest point in a convex set, analogous to finding the lowest or highest elevation on a smooth, rounded hill.

Critical Features of Convexity

Some advantages of convexity in optimization include the guarantee of local minima being global minima and the relative ease of analyzing and computing solutions. When dealing with a convex problem, the journey down the 'hill' is straightforward: every step downhill moves you closer to the lowest point, with no risk of falling into a 'lower valley' – a property not found in non-convex problems.

Hence, convexity is highly valued in optimization for both its theoretical elegance and its practical implications — it assures that our search for an optimal solution is predictable and less prone to complications that arise from the landscapes of more complex shapes.