The optimal gantry shape may be a triangle. peteeng has posted some good information and simulations on triangular shaped gantry beams.
The real test is can a good triangular gantry be practically built (and beat the alternatives) by / for the hobbyist.
Basics:
Steel appears to be the best material. Steel being very stiff, easily laser cut, and relatively cheap.
Laser cutting (or water jet, or plasma cutting) is likely optimal, being reasonably affordable and precise
Fabrication
How do we stick the components together to form the gantry?
Welding
Pros:
Readily available
Low entry cost
Can fill loose joints.
Cons:
Distortion, distortion, distortion
Increases residual stress -> stress relief optimal / may be required -> added $$$
Brazing
Sounds great in theory - low heat, so does not cause an increase in material stress
Can joint dissimilar metals
BUT requires tight fit-up and good cleaning.
Most sources quote an optimal joint space ("fit up") of 0.003-0.006" (0.0762 - 0.1524mm) for maximum strength (looser tolerance is possible, but strength falls off and chance of a bad joint increases)
Cleaning flat exposed surfaces is not hard, but cleaning the inside of laser cut slots (with width = material thickness) may be difficult (and may affect fit-up if using abrasive).
The need for tight fit-up is a major limitation in this application where 45º joints are involved. Almost all laser cutting / waterjet / plasma will be on 3 axis machines - thus a 45º chamfer edge will not be produced by these machines. Adding this chamfer whilst retaining tight tolerance appears unlikely for the hobbyist
Glue / epoxy
Low stiffness
Optimizing the shape
A 90 degree right angle triangle with 45º angles seems optimal and easiest for most situations.
Linear bearings on the vertical face - both on the same face makes machining and aligning much easier
Vertical ribs - spacing of ribs etc will depend a lot of method of fabrication - need to access each joint in turn.