Pronunciation: [kɒnvˈɛksɪz] (IPA) 
The spelling of the word "convexes" can be pronounced as /kənˈvɛksɪz/. The word is a plural form of "convex", which refers to a curved surface that bulges outwards, such as a convex mirror. The "-es" suffix is added to the base word to indicate pluralization. The pronunciation of the word follows the English phonetic rules, where "c" is pronounced as /k/, "e" as /ɛ/, and "s" as /s/.
Convexes, also known as convex sets, are mathematical entities that have specific properties in the field of geometry. A convex set is defined as a subset of a vector space, typically in Euclidean space, where for any two points within the set, the line segment connecting those points lies wholly within the set itself.
In simpler terms, a convex set can be visualized as a shape that is "curving outwards", bulging or rounding away from its interior. This curvature property is such that if you were to draw a straight line connecting any two points within the set, that line would never cross or extend beyond the set's boundary.
Examples of convex sets include a circle, a triangle, a square, or even the entire Euclidean space itself. On the other hand, shapes like a crescent, a hollow square with a hole in the center, or a horseshoe cannot be considered convex sets, as there are points where the line connecting them would exit the shape.
Convex sets are widely studied and applied in various fields, such as optimization, economics, computer science, and physics. Their properties, such as being closed under convex combinations and their significance in optimization problems, make convex sets an essential concept in mathematics.
The word "convexes" is the plural form of the noun "convex". The etymology of "convex" stems from the Latin word "convexus", which means "arched" or "rounded". This Latin term is derived from the prefix "co-" (meaning "together") and the root "vexus" (meaning "bent" or "curved"). Over time, the word "convexus" evolved into the English word "convex" to describe a surface that bulges outward, being curved or rounded like the exterior of a sphere. By adding the plural suffix "-es" to "convex", we get "convexes" to refer to multiple convex objects or surfaces.
