You have a radially sawn slab of wood, indicated by the visible medullary rays on both faces, to which has been attached mouldings to one face. The most likely scenario is that the slab would have been flat at the time of manufacture and there has been shrinkage across the width of the slab caused by moisture loss probably fairly soon after manufacture and installation in someone's residence. Normally, because the wood slab is quarter (radially) sawn it would be relatively stable during cross grain shrinkage and remain pretty flat without the visible cupping. The cupping is almost certainly caused by the two pieces of cross grain moulding applied to the show face.
During shrinkage the cross grain mouldings won't shrink with the slab beneath it so it has three options:
1. Remain firmly attached (glued, pinned, screwed[?]) and push at the mitred intersections where the long mouldings meet the cross grain mouldings. This would cause the meeting at the mitred point to become misaligned and perhaps result in a gap or a step(s).
2. The cross grain mouldings and/or the long edge mouldings lose their firm attachment, at least in part, again resulting in gaps or steps at places like the mitres and along the length of one or more mouldings.
3. The cross grain mouldings remain firmly attached as at 1 above but as the slab shrinks there is little or no effect at the corner mitres (misalignment), or gappiness, etc elsewhere, and because of differential shrinkage between the slab and the mouldings, i.e., the slab shrinks but the mouldings don't the slab has to cup convex on the show face and the mouldings curve with the cup.
There isn't a simple fix using moisture, steam, cramps, pressure, etc. The only fix is disassembly, flatten the slab and reattach the mouldings which will almost certainly mean shortening the length of the cross grain mouldings be a millimetre or two. It sounds fairly straightforward, and it is if you know what you're doing, but may also be relatively time consuming and almost certainly would involve at least some stripping of polish and refinishing to match. Add to that, fix may only be temporary because the slab beneath the mouldings may want to expand or contract again due to a change in the relative humidity range the piece of furniture experiences in the future.
Your option of "embracing the curve and resetting the hinges to compensate" might be your best choice, and depends upon the skills you have to draw upon. Slainte.
