The manufacturing itself won't be cheaper. But you can save transport costs. It costs about 22000 USD to send 1 kg into space. SpaceX might improve this to 1000 USD per kg, but that's still quite expensive.

Manufacturing and processing materials to sustain martian colonies, exploration, or satellites. The cost will be high initially, but if we're ever to have any permanent foothold, local manufacturing will be needed.

Not necessarily on Mars. Earth orbit would be a good start, so would be Luna if you absolutely need to have a solid rock bed under you.

It definitely won't be cheaper initially, especially if you want that manufacturing to serve the needs of people down on Earth; but it will make more space development cheaper (you don't have to move mass upwell, and as we know, Low Earth Orbit is halfway to anywhere in the Solar System). So space industry will most likely initially work on developing itself and reducing its own costs.

One the arguments Bezos floats for space-based manufacturing is to offplanet the negative environmental consequences of industry. His premise is, the earth is for habitation, space is for industry. Certainly seems likely to be very expensive, but he also thinks he can put down a foundation of space infrastructure upon which the next generations can build upon (with the implication being to bring costs down dramatically to enable a boom in all things space). He uses the Internet as a comparison, in that he was able to do what he did, because previous generations put down a technology foundation that made Amazon possible (he didn't have to build all of that).

There's abundant calcium, silicon, aluminum, titanium and oxygen on the moon. You bring in hydrogen and export tritium and a lot of the rest could be used to expand settlements and build up space infrastructure.

Maybe not on Mars, but in low gravity environments. Metallurgy in low-gravity is interesting.

That works for the ISS, maybe a DC would just be bigger in scale.

There is also the link problem: if the space DC is geostationary, it will be quite far making for a longer round-trip time (RTT) and latency. If it's closer, it needs a lot of radio stations on earth to keep the space DC connected. And each radio station is roughtly the investment of a real DC on Earth.

Maybe you plant your data center on the dark side of an iron-rich asteroid and sink waste heat into it.

It always is :)

Tens of kilometres of hard vacuum too..

Microwaves. The problem with that though is, from what I hear, such technology would double as an orbital weapons platform.

I wonder what the math says about big mirrors focusing more sunlight over empty areas where we could put lots of solar panels.