How can I define one path for multiple shapes from either side of it in Illustrator?

Question

New to Illustrator, I have trouble solving this.

I have an image (whose paths you can see below in the image marked in different colors), and I want to create paths for the shapes inside the existing paths, i.e. the white faces. What I need in the end is an SVG which will contain just these inner paths for the white shapes.

I tried grouping different paths, but it seems that Illustrator allows one path to be part of only one group. I tried just selecting them and creating live paint groups but that fails too because I get weird inner shapes formed by the vertices and plus I'm not sure if the end result will be the inner paths I need.

So, how do I approach this?

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the svg file - paths in different colors bound the surfaces i want to create paths for

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The paths that form the grayed out areas is what I need

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Answer 1

The Live Paint function is probably what you need, although your question is a bit unclear, I think this is what you want to do:

Once you have created your first strokes, select them all and click Object > Expand to turn them into outlines rather than stokes. And then open the Pathfinder, and Unify all the outlines.

Then turn them into a live paint object, fill the spaces with the Live Paint Bucket. Select the Live Paint object, then click Object > Expand, and then Object > Ungroup (do this twice). Select the black lines, and delete them, leaving only the inside outlines.