f-vector

Builds on Convex polytope Face of a convex set Simplicial complex

Definition

Let $P$ be a convex polytope of dimension $d \ge 1$ . For $0 \le i \le d - 1$ , let $f_i(P)$ be the number of faces of $P$ of dimension $i$ . The f-vector of $P$ is $(f_0(P), f_1(P), \dots, f_{d-1}(P))$ .

Remarks

Some authors add $f_{-1} = 1$ for the empty face and $f_d = 1$ for $P$ itself. For a finite simplicial complex $\Delta$ , $f_i(\Delta)$ is the number of $i$ -dimensional simplices of $\Delta$ , defined in the same way.

Example: the cube $[0,1]^3$ has $8$ vertices, $12$ edges and $6$ square faces, so its f-vector is $(8, 12, 6)$ .

Non-example: $(4, 6, 3)$ is not the f-vector of any $3$ -dimensional polytope. Every $3$ -polytope satisfies $f_0 - f_1 + f_2 = 2$ , while $4 - 6 + 3 = 1$ .

Reading path

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Convex polytope concept
Face of a convex set concept
Simplicial complex concept
f-vector definition

Tag	`SJDZH6V0BC`
URI	`m2://SJDZH6V0BC`
Kind	definition
Writing	W0 stub
MSC	52B05, 05E45
Defines	f-vector
Uses	Convex polytope, Face of a convex set, Simplicial complex

f-vector

Definition

Remarks

Reading path

Further reading

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