You did not give an example function, so I am making one up; say your "function with a peak" is:
g[x_, y_] := 1/(x^2 + 2 y^2)
and you want to know the dimensions at g=1, then this is maybe giving what you are asking for:
reg = ImplicitRegion[g[x, y] > 1, {x, y}];
CoordinateBounds[reg]
