The lower envelope is the boundary you see when looking at all lines from below. It's a piecewise linear function: each segment belongs to one line. If you sort lines by slope, the lower envelope forms a convex shape (curving outward like the bottom of a bowl).
That's why this technique is called the "Convex Hull Trick". Lines appear left-to-right in slope order. Adjacent lines on the envelope intersect at breakpoints. For query , find which segment contains . That segment's line gives the minimum. The structure has at most segments.