Overview
Important
Convex analysis studies shapes (sets) where, if you pick any two points inside, the straight line between them stays inside the shape. Linear programming is a method to find the best value (maximum or minimum) of a linear function, subject to linear constraints, often visualized as a region on the plane or in space.
Important properties
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A set is convex if, for any two points inside it, the segment joining them is also inside.
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The feasible region of a linear programming problem is always convex.
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The optimal value of a linear programming problem (if it exists) is always achieved at a vertex (corner point) of the feasible region.