AMU 2026 — Mathematics PYQ

In an LPP, the constraints are defined by linear inequalities. When these inequalities are plotted on a graph, they define a region representing all possible solutions that satisfy every constraint simultaneously. This is called the feasible region.

1. Why it is Convex:

   A set is defined as convex if, for any two points $P_1$ and $P_2$ within the region, the entire line segment connecting them also lies entirely within that region. Because all LPP constraints are linear (represented by straight lines or planes), the intersection of these half-spaces always results in a convex polygon (in 2D) or a convex polyhedron (in 3D).

2. Mathematical Representation:

   If $X_1$ and $X_2$ are two feasible solutions in the region, any convex combination of these points, defined as:

   $$\lambda X_1+(1-\lambda )X_2,\text{where }0\le \lambda \le 1$$

   will also be a feasible solution.

Because the feasible region is formed by the intersection of linear inequalities, it cannot be concave. A concave region would contain "dents" or inward curves that violate the property of linear constraints.