= Capped simplex
For an integer $0\leq k\leq n$, the capped simplex
$$
P_k=\{v\in\mathbb R^n:0\leq v_i\leq1,\ \sum_i v_i=k\}
$$
is the <convex hull> of the zero-one <vectors> having exactly $k$ 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 $[0,1]^n$, since each coordinate is already at a bound.
Back to article page
