A convex combination is a weighted sum of points where all individual weights are non-negative numbers adding up to one, and its weight distribution structurally defines the interior and boundary of a feasible region. The interior consists of points where every single weight is strictly greater than zero, pulling the point inward away from all bounding walls so that no constraints are binding. Conversely, the boundary is formed when one or more weights drop to zero, restricting the combination to lower-dimensional edges or vertices and pushing the point outward onto the hyperplanes where at least one constraint holds with strict equality.

Points in the interior are formed using strictly positive weights across all defining vertices, placing them strictly inside the feasible region. Points on the boundary occur when one or more weights are set to zero, restricting the combination to the region's outer edges or faces.

A convex combination creates points between feasible points. Boundary points lie on the edge, while interior points lie inside the feasible region.

A convex combination shows that points inside the feasible region can be formed between feasible points, while points on the boundary lie at the edges of this region.

A convex combination expresses points between feasible points: points formed using positive weights summing to 1 lie in the interior, while combinations reaching the limits of the feasible set lie on its boundary.

How does a convex combination help determine whether a point lies on the boundary or in the interior of a feasible region?