For finite-dimensional $V$, your definition of a face is equivalent to the definition of a poonem, according to part (i) of Exercise 7 on page 21 of the book Gruenbaum; then the positive answer to your question is part (iii) of Exercise 3 on the same page.
Iosif Pinelis
- 127.9k
- 8
- 107
- 229