Remove legend if a plot in an array of plots is empty

Question

In an array of plots of regions (inequalities) p[..] which are eventually combined using Show, how do I suppress the legend of plots which are empty, and look like this

More generally, if in a given range of $x$ and $y$ over which a region is prescribed to be plotted, if RegionPlot does not find a solution (so that there is no region to shade or fill), is there a way to get a True or False value from RegionPlot corresponding to whether the region exists or not?

Answer 1

You can express the region using ImplicitRegion and then plot it or check the area of the region. Example:

reg = ImplicitRegion[x^2 + y^3 < 2, {{x, -2, 2}, {y, -2, 2}}];
RegionPlot[reg]

Chop@N@Area[reg]

9.91915

Checking to see if the area of the region is nonzero:

If[Chop@N@Area[reg] > 0, ...]

Answer 2

colors = RotateLeft[ColorData[97] /@ {1, 2, 3}];
regs = {x^2 < (y - 2)^3 + 1, (y + 2)^2 < (x - 2)^3 + 1, x + y <= -2}; 

Show[With[{col = First[colors = RotateRight[colors]]}, 
    RegionPlot[#, {x, -5, 5}, {y, -5, 5}, PlotStyle -> col, 
     PlotLegends -> If[Area[ImplicitRegion[#, {{x, -5, 5}, {y, -5, 5}}]] === 0, None, {#}], 
     PlotRange -> {{-5, 5}, {-5, 5}}]] & /@ (And[#, True] & /@ regs)]

Replace And[#, True] & with And[#, x >= 2 y] & to get

Replace And[#, True] & with And[#, x <= y] & to get