Insulating a shallow exterior wall?

Don’t compress any insulation, ever. The glass fibers in the pink stuff is highly conductive of heat, it is the air trapped inside that provides the insulation. The reason you put the glass fibers in the space rather than just leaving an air filled cavity is because convection currents can form inside the wall if you have cavity which is 2 inches or thicker. This convection current will carry cool air to the warm side and warm air to the cool side of the cavity very quickly.

If the thickness of the cavity is only 1 inch, just leave the space empty, but make sure you seal any holes (think electrical wiring/outlets). Convection currents won’t start up in such a thin space, there will be turbulent flow, but a thin air filled cavity with no holes for infiltration will give a better R value than any insulation material you install. Double and triple paned glass windows work well because the space between the panes is way too thin for a convection current to form.

There are three ways heat moves through the walls of a house. Direct conduction: this is mostly a problem with metal window frames and any single paned glass windows, if you have double paned glass in wood or plastic frames and live in a brick house this is the least problematic of the three. Radiation: this is mostly a problem of windows and mostly a summer problem, in the summer infrared light passes in through the windows, and in the winter it passes out through the windows. This can be addressed with a thin plastic coating on the window that is opaque to infrared light; usually you also get a second plastic coating for UV as well, which will protect your books and fabrics inside but is completely unrelated to the infrared blocking ability of the window; at the moment these windows are too expensive to be cost effective, unless you need the UV protection and this is just a bonus coating. The most heat is transferred by the movement of air. You can address this by closing holes, sealing up electrical boxes, maintain the chaulking in your house and sealing any air leaks you can find (under the door, always open windows, always open vents, etc.) You can use incense to generate smoke and some fans to generate positive pressure in the house, then the smoke will travel straight to the holes..

PS You can do better than an empty cavity for a thin space like this. There are wonder materials such as aerogels and certain gas filled ceramic tiles that are absolutely amazing in their thermal properties. But they would costs millions of dollars to install in a 10’x10’x1″ wall so they will not be practical for home improvement until all the patents expire. You could also find a thin sheet of insulation, but for such a thin space and a brick exterior wall I doubt it is worth it.

PS2 I’m assuming you are in a brick house, if you are in a wood frame house then you can transfer large amounts of heat through the walls because the inner and outer membranes are much thinner and the materials are more dense than brick, which means they are highly thermally conductive. The air space in-between the hightly conductive inside wall and the hightly conductive outside wall needs to be packed with some low density material to prevent a convection current from forming. Once a current forms inside the wall it is as if your walls were just one single thin highly conductive material, like a single paned window (except without being transparent to infrared radiation).

my understandiing (what we are doing) is that you should always maintain an airspace inside of the brick to allow the brick to breath. In our space constrained walls, we have used 2″ thick foam board separated from the brick by the furring strps but it sounds like this is still to thick. Probably your best best is to just use a reflective foil vapor barrier which increases your r value (I believe particularly with radiant heat systems) while taking up negligible space. I wouldn’t fill the space with foam nor with any bat insulation.