If S is a polytope, then the k-extreme points are exactly the interior points of the k-dimensional faces of S. More generally, for any convex set S, the k-extreme points are partitioned into k-dimensional open faces.

If the object is the union of a finite set of points, its convex hull is a polytope.

If the objective function f is linear and the constrained space is a polytope, the problem is a linear programming problem, which may be solved using well known linear programming solutions.

If a supporting hyperplane also intersects the polytope, it is called a bounding hyperplane ( since it is a supporting hyperplane, it can only intersect the polytope at the polytope's boundary ).

If it is not, then the solution of Ax ≤ b lies in a proper affine subspace of R < sup > n </ sup > and discussion of the polytope may be constrained to this subspace.

If a polytope is d-dimensional, its facets are its ( d − 1 )- dimensional faces, its vertices are its 0-dimensional faces, its edges are its 1-dimensional faces, and its ridges are its ( d − 2 )- dimensional faces.

# If r is integral, 1-Lipschitz, and then r is the rank-function of a matroid, and the polymatroid is the independent set polytope, so-called since Edmonds showed it is the convex hull of the characteristic vectors of all independent sets of the matroid.

If a broader Atlantic community is to be formed -- and my own judgment is that it lies within the realm of both our needs and our capacity -- a ready nucleus of machinery is at hand in the NATO alliance.

If our national interest lies in being able to fight and win a war rather than committing national suicide, then we must take a much more penetrating look at ballistic missiles.

`` If Philip Toynbee is claiming that the choice lies between capitulation and the risk of nuclear war, I think he is right.

If the stud attempts to mate with her and she lies down for him within a fairly short period of time, she is not pregnant.

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean.

If it lies around the foundations it may cause seepage to the interior, rising damp or dry rot.

If the axis of the rotation lies external of the body in question then the body is said to orbit.

If it works, the problem being diagnosed likely lies in the real application being diagnosed.

If its density is similar to Amalthea's, Metis's orbit lies within the fluid Roche limit ; however, since it has not broken up, it must lie outside its rigid Roche limit.

If, for a destination, a neighbor router advertises a distance that is strictly lower than our feasible distance, then this neighbor lies on a loop-free route to this destination.

If, for a destination, a neighbor router tells us that it is closer to the destination than we have ever been, then this neighbor lies on a loop-free route to this destination.

If the value of this is negative then the midpoint lies above the line, and if the value of this is positive then the midpoint lies below the line.

If the midpoint lies below the line then should be chosen since it is closer ; otherwise should be chosen.

## If that polygon lies in the plane containing P, add it to the list of polygons at node N.

If you can't handle the thought that the President lies to the public for all kinds of reasons, you couldn ’ t stay in the government at that level, or you ’ re made aware of it, a week.

If three lies were spoken over it, it would break in three ; three truths made it whole again.

If origin is equivalent to power, then " it is the first manifestation of a thing that is significant and valid " ( a thing's reality and value therefore lies only in its first appearance ).

If they do not return to the main party within one week with news of what lies beyond, they will be considered lost.

If the patient lies down, the symptoms usually vanish in a short amount of time.

If retreating underground is not possible, she collects all young together and lies on top of them.

If the object lies outside the Earth's atmosphere, as in the case of stars and planets, the phenomenon is termed astronomical scintillation ; if the luminous source lies within the atmosphere, the phenomenon is termed terrestrial scintillation.

If Russian pupils have to take these languages, how come American students have a choice whether or not to take a language, but have to face so many exceptions??

If, after signing such a declaration, the information is found to be deliberately untrue with the intent to deceive, the applicant may face perjury charges.

In the summer of 1862 he said, " If you persist in forcing the issue of slavery against the government, I say in the face of heaven: Give me my government and let the negroes go.

If anyone at my funeral has a long face, I'll never speak to him again.

If peace were made, the armies would return home and the directors would have to face the exasperation of the rank-and-file who had lost their livelihood, as well as the ambition of generals who could, in a moment, brush them aside.

If Language A borrowed a word from Language B, or both borrowed the word from a third language or inherited it from a common ancestor, and later the word shifted in meaning or acquired additional meanings in at least one of these languages, a native speaker of one language will face a false friend when learning the other.

If Hezekiah expected the Egyptians to come to his aid, they did not come, and Hezekiah had to face the invasion of Judah by Sennacherib () in the 4th year of Sennacherib ( 701 BC ).

If any other person has put in the eye and eyebrow of a face, I can perceive whose work the original face is, and who has painted the eye and eyebrow.

If the accusers were among those named, the defendant was set free and the charges dismissed ; the accusers would face life imprisonment.

If the case has been submitted to arbitration rather than a national court, say because of a forum selection clause, an arbitrator may decide not to apply local mandatory policies in the face of a choice of law by the parties if this would defeat their commercial objectives.

Straight or Flush with one pair: If you can play a face pair with an ace-face top when the straight or flush gives only a Q-x or lower top, play the strong pair with an ace-face top.

If however the reverse waist lock is set from the back of the opponent, then the latter would face away from the athlete in the inverted position.

If the visitor fails to do so, usually a quarantine fine of 220 Australian dollars are to be paid as quarantine infringement notice, and if not, the visitor may face criminal convictions of fining 100, 000 Australian dollars and 10 years imprisonment.

The passers-by spit in his face, and sometimes beat him … unmercifully … If a Jew enters a shop for anything, he is forbidden to inspect the goods … Should his hand incautiously touch the goods, he must take them at any price the seller chooses to ask for them ...

If one were to only take what was at face value observable, many scientific phenomena never would have been discovered.

* If it was an abortion, usually measured after three months in a pregnancy, the woman may pay 1 / 10 of the blood money to charity or face imprisonment ( not qisas ).

If the male is an Abdi Dalem royal servant, courtier or particularly " peko-peko " ( taken directly from Japanese to mean obsequious ) or even a highly formal individual, he will retreat face and head downcast, never show his side or back to his superior, and retreat backwards in the following posture: left-arm crossed against the chest, right-arm hanging and walking stooped.

If an unbroken glyph is used, then the hand is placed in the vertical ( wall or face ) plane in front of the signer, as occurs when finger spelling.

If it chooses the former, the telephone company will face less regulation but will also have to comply with FCC regulations requiring what the Act refers to as " open video systems.

If the end face of the fiber is perpendicular to the fiber axis, the launch angle is equal to the angle of incidence.

If " good " coins have a face value below that of their metallic content, individuals may be motivated to melt them down and sell the metal for its higher intrinsic value, even if such destruction is illegal.

Henny, who got an apple in his face, responded by saying " If you keep this up then there'll be a real Nightnurse on hand.

If not, they face disqualification.

If the problem resides in the facial nerve itself ( peripheral palsy ) all nerve signals are lost on the ipsilateral ( same side of the lesion ) half side of the face, including to the forehead ( contralateral forehead still wrinkles ).