tl;dr: “Virtual space” is a poor metaphor, we shouldn’t take it too literally, especially when hypothesizing about the legal ramifications of putting things “in it”. 

The topic of who owns virtual space (in front of existing advertising billboards outdoors, in museums and other curated spaces) comes up often in discussions about AR and its implications, and for me this is a symptom of people taking AR metaphors too literally. When we imagine physical spaces as if they have a single contiguous virtual volume, we can end up projecting the limitations of physical space onto digital data. This is unnecessary, inappropriate, and leads to confusion.

What’s actually happening with (visual) AR? Well, we have a view on the world, and we have some geo-spatial data comprising a thing to render, an understanding of how to render it, and metadata describing where in a coordinate frame it is addressed. At some point, we composite a suitable rendering of that data with a view of the real environment (our actual view, a video stream, etc). Essentially we are using a spatial descriptor, be it a pose reading from sensors, or feature points from a camera image, to generate an address in a data representation scheme, and we’re fetching data proximal to that address in a given database. We may “align” the rendering with the view, but this happens in 2D at the rendering stage.

The geo-spatial data is infinite: we can create any number of database instances that will return any number of different data objects for the same coordinate. Equally, creating an geo-located AR 3D model doesn’t “use-up” “virtual space” that could be occupied by someone else, and you can have multiple objects associated with a given coordinate. Understanding this makes the idea of geo-spatial digital property claims ridiculous (that over 2000 potentially coincident AR layers happily coexist simultaneously on the same platform drives home the point). So even if we hang on to “virtual space”, even just as a metaphor, “virtual spatial layers” is better - it emphasizes the multiplicity (even within platforms). 

Examining one of the excellent hypothetical questions:

So what would happen if someone painted an AR beard on the Mona Lisa

The beard would reside not in the space between you and the Mona List, but as a data entry in a particular database, run by a particular service, viewable with a particular application (today). Nothing happens, but it becomes possible to find and view the AR beard if so desired. The process of rendering the beard happens entirely on your device (whatever it may be). And as discussed previously, the viewing is inherently opt-in. 

Considered like this, the legal issues look more likely to surround copyright and the combination of the virtual overlay and the overlaid taken together as comprising a derivative work than geospatial positioning. Copyright issues are likely to be irrelevant in public commons, and in cases where there is no copyrighted object being derived from (e.g. if you place your AR piece in an empty piece of museum). (Of course, IANAL).

So, AR metaphors shouldn’t be taken too literally, and it would be much better to promote non-metaphorical thinking about AR when considering legal ramifications even hypothetically, to try to limit the scope to well-defined issues like copyright and derivative works, rather than to promote “virtual space” thinking which is liable to cloud the issues and potentially lead to much more restrictive and insane legislation like taking ownership of ranges of addressing schemes that correspond to physical volumes, or partitioning ranges to different AR vendors etc. 

That said I’d really like to get other people’s take on this.