For an integer , the capped simplexis the convex hull of the zero-one vectors having exactly entries equal to one. It is a convex polytope whose extreme points are precisely these vectors.
If a feasible vector has a fractional coordinate, the integer coordinate sum forces at least two fractional coordinates. Perturbing one upward and the other downward by a sufficiently small amount gives two distinct feasible vectors whose midpoint is the original vector. It is therefore not an extreme point. Conversely, a zero-one vector cannot be a nontrivial convex combination of points of , since each coordinate is already at a bound.
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