Your definition (if made precise) is quite long, and it does not worth to define polyhedral space this way.

You may do the following two things:

1. Define it as a finite (or locally finite) simplicial complex with a length metric such that each simplex is isometric to Euclidean simplex.

2. Define it as a compact (or locally compact) metric space such that each vertex adimits a cone neighborhood.

The second definition is weaker than yours and the first is stronger. The equivalence of these two definitions is written, see Local characterization of polyhedral spaces by Lebedeva and me.

Your definition (if made precise) is quite long, and it does not worth to define polyhedral space this way.

You may do the following two things:

1. Define it as a finite (or locally finite) simplicial complex with a length metric such that each simplex is isometric to Euclidean simplex.

2. Define it as a compact (or locally compact) metric space such that each vertex adimits a cone neighborhood.

The second definition is weaker than yours and the first is stronger. The equivalence of these two definitions is written, see Local characterization of polyhedral spaces by Lebedeva and me.

Your definition (if made precise) is quite long, and it does not worth to define polyhedral space this way.

You may do the following two things:

1. Define it as a simplicial complex with a length metric such that each simplex is isometric to Euclidean simplex.

2. Define it as a compact (or locally compact) metric space such that each vertex adimits a cone neighborhood.

The second definition is weaker than yours and the first is stronger. The equivalence of these two definitions (at least in compact case) is written, see Proposition 2.12 Lebedeva's paper.

Your definition (if made precise) is quite long, and it does not worth to define polyhedral space this way.

You may do the following two things:

1. Define it as a finite (or locally finite) simplicial complex with a length metric such that each simplex is isometric to Euclidean simplex.

2. Define it as a compact (or locally compact) metric space such that each vertex adimits a cone neighborhood.

The second definition is weaker than yours and the first is stronger. The equivalence of these two definitions is written, see Local characterization of polyhedral spaces by Lebedeva and me.