There’s a swing arm halogen lamp (kind of like this) I use for lighting eBay photos that’s taken a few tumbles to the floor over it’s service life. I touched the swing arm section of the lamp right where it pivots about 18 inches from the bulb head section and the metal was very hot. Not red hot, but way too hot to keep touching it more than a second or so.
Is this an electrical short of some kind or is this simply a characteristic of halogen lamps?
There may be a pivoting electrical contact at the joint - if so, it probably has a high resistance due to wear. Probably not too dangerous, but if you took it apart and cleaned it, it would make the light brighter and the joint cooler.
Halogen lamps do indeed get very hot. Usually there’s a knob or lever that’s available to angle the light to where you want it. This knob or lever is generally insulated enough so that it’s not hot to the touch. Maybe it broke off without your realizing it. I doubt that you have a short in the lamp. It’s probably just heat traveling down from the bulb area. I’d suggest just wrapping some insulation around the part where you normally touch the lamp.