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I have a certain three-dimensional constraint

-(1/3) < t3 < 1/3 && -(1/3) < t2 < 1/3 && 9 t1^2 < (1 - 3 t3)^2 && 9 t1^2 < (1 + 3 t3)^2 && t1^2 t2^2 t3^2 > 1/23328  .

If I perform an integration over the unit cube $[-1,1]^3$, I obtain (using the GenericCylindricalDecomposition and FullSimplify command) the result

(1/(54 Sqrt[2]))(8 Sqrt[2 - Sqrt[2]] - ArcTanh[Sqrt[1 - 1/Sqrt[2]]] (8 + Log[128]) + Log[2 - Sqrt[4 - 2 Sqrt[2]]]^2 - Log[2 + Sqrt[4 - 2 Sqrt[2]]]^2 - 4 PolyLog[2, 1/4 (2 - Sqrt[4 - 2 Sqrt[2]])] + 4 PolyLog[2, 1/4 (2 + Sqrt[4 - 2 Sqrt[2]])]

which evaluates to approximately 0.00221357, while, of course, the unit cube has a much larger volume, that is, $2^3=8$.

I believe--on the basis of prior ("bound-entanglement" probability) considerations, that the regions in which the constraint is satisfied are disjointed and form an "archipelago" of islands.

However, my limited attempts (using various ranges of coordinates for the variables $t1, t2, t3$) to show these presumed regions using RegionPlot3D just return vacuous plots.

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RegionPlot3D missed the regions of interest when using the default number of plot points because they are a very small fraction of the entire plotting volume. You can increase the PlotPoints used to get a result:

RegionPlot3D[
  -(1/3) < t3 < 1/3 && -(1/3) < t2 < 1/3 && 9 t1^2 < (1 - 3 t3)^2 && 
    9 t1^2 < (1 + 3 t3)^2 && t1^2 t2^2 t3^2 > 1/23328,
  {t1, -0.6, 0.6}, {t2, -0.6, 0.6}, {t3, -0.6, 0.6},
  PlotPoints -> 200, Mesh -> None
]

3Dplot with restricted range

Note that I restricted the plotting range to emphasize the regions of interest. You can use the full $(-1,1)$ range and it works the same. The Mesh -> None option is simply to produce a more readable plot, but it has no functional effect.

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  • $\begingroup$ I've tried upping the number of PlotPoints--but the resulting plots seem to stay exactly the same. I strongly suspect that these islands have a craggy/rough nature. Is it possible to reveal that as well? As it stands, they seem to have a rather smooth character. Maybe one would have to zero in on particular islands to achieve that. $\endgroup$ – Paul B. Slater Dec 18 '19 at 20:11
  • $\begingroup$ @Paul I haven't zoomed very far in, but from a quick glance at the result of Reduce[yourconditions, {t1, t2, t3}] the boundaries of these lunettes seem to be fourth-order polynomials, so they may actually be relatively smooth. $\endgroup$ – MarcoB Dec 18 '19 at 23:14

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