Which inequality defines the region of interest in the hyperclip formulation?

Which inequality defines the region of interest in the hyperclip formulation?

• A. A^T X ≥ 0
• B. A^T X + R ≤ 0
• C. X^T A + R ≥ 0
• D. A X + R ≤ 0

Answer: (B)

The region of interest is defined by a collection of linear inequality constraints that carve out a clipped region in space. In the hyperclip formulation, each constraint has the form A^T X + R ≤ 0, meaning for every row of A, the dot product with X plus a corresponding offset must be at most zero. Geometrically, each constraint defines a half-space bounded by the hyperplane a_i^T X = -r_i, and taking the intersection of all these half-spaces gives the clipped region where X is allowed to lie. This form correctly captures the idea of clipping by margins: the offsets in R shift the boundaries, so only points that satisfy all shifted half-spaces are inside. If any constraint is violated, the point is outside the region. The other expressions don’t match this clipping structure. Using A^T X ≥ 0 removes the offsets and selects the opposite side of the boundary. Writing X^T A + R ≥ 0 alters the orientation and, depending on dimensions, isn’t the same linear constraint as A^T X + R ≤ 0. Using A X + R ≤ 0 multiplies X by A in a way that changes the geometry (and can be dimensionally inconsistent) and does not represent the same set of clipped points.

The region of interest is defined by a collection of linear inequality constraints that carve out a clipped region in space. In the hyperclip formulation, each constraint has the form A^T X + R ≤ 0, meaning for every row of A, the dot product with X plus a corresponding offset must be at most zero. Geometrically, each constraint defines a half-space bounded by the hyperplane a_i^T X = -r_i, and taking the intersection of all these half-spaces gives the clipped region where X is allowed to lie.

This form correctly captures the idea of clipping by margins: the offsets in R shift the boundaries, so only points that satisfy all shifted half-spaces are inside. If any constraint is violated, the point is outside the region.

The other expressions don’t match this clipping structure. Using A^T X ≥ 0 removes the offsets and selects the opposite side of the boundary. Writing X^T A + R ≥ 0 alters the orientation and, depending on dimensions, isn’t the same linear constraint as A^T X + R ≤ 0. Using A X + R ≤ 0 multiplies X by A in a way that changes the geometry (and can be dimensionally inconsistent) and does not represent the same set of clipped points.