Convex Analysis is the calculus of inequalities while
Convex Optimization is its application. 
Analysis is inherently the domain of the mathematician while Convex Optimization belongs to the engineer.

Philip Gill,  Walter Murray,  Michael Saunders,  Margaret Wright
Pioneers of Mathematical Optimization.

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In layman's terms, the mathematical science of Convex Optimization is the study of how to make a good choice when confronted with conflicting requirements.  The qualifier convex means: when an optimal solution is found, then it is guaranteed to be a best solution; there is no better choice.

Convex Optimization & Euclidean Distance Geometry is about convex optimization, convex geometry (with particular attention to distance geometry), and nonconvex, combinatorial, and geometrical problems that can be relaxed or transformed into convex optimization problems.

Existence of geometric interpretation for any convex optimization problem is a certainty.  If a problem can be transformed to an equivalent convex optimization, then ability to visualize its geometry is acquired. 

There is a great race under way to determine which important problems can be posed in a convex setting.  Yet, that skill acquired by understanding the geometry and application of convex optimization will remain more an art for some time to come; the reason being, there is generally no unique transformation of a given problem to its convex equivalent.  This means, two researchers pondering the same problem are likely to formulate the equivalent differently; hence, one solution is likely different from the other for the same problem.  Any presumption of only one right or correct solution becomes nebulous.  Study of equivalence, sameness, and uniqueness therefore pervade study of convex optimization.