`FindInstance[ 0 < x1 + x2 + x3=x4 < 2 && 0 < x4 < x3 < x2 < x1 < 1 && 1/x1 + 1/(x1 - x2) + 1/(x1 - x3) + 1/(x1 - x4) < 1/(1 - x1) && 1/x2 + -1/(x1 - x2) + 1/(x2 - x3) + 1/(x2 - x4) < 1/(1 - x2) && 1/x3 - 1/(x1 - x3) - 1/(x2 - x3) + 1/(x3 - x4) < 1/(1 - x3) && 1/x4 - 1/(x1 - x4) - 1/(x2 - x4) - 1/(x3 - x4) < 1/(1 - x4), {x1, x2, x3, x4}, Reals] FindInstance[ 2 < x1 + x2 + x3+x4 < 2.1 && 0 < x4 < x3 < x2 < x1 < 1 && 1/x1 + 1/(x1 - x2) + 1/(x1 - x3) + 1/(x1 - x4) < 1/(1 - x1) && 1/x2 + -1/(x1 - x2) + 1/(x2 - x3) + 1/(x2 - x4) < 1/(1 - x2) && 1/x3 - 1/(x1 - x3) - 1/(x2 - x3) + 1/(x3 - x4) < 1/(1 - x3) && 1/x4 - 1/(x1 - x4) - 1/(x2 - x4) - 1/(x3 - x4) < 1/(1 - x4), {x1, x2, x3, x4}, Reals] `

I’m trying to run the above above commands. I suspect that the first wouldn’t have a solution while the second would, however FindInstance does not return an answer, or at least running it for an hour didn’t result in anything. How can I speed up FindInstance, at least in this case?

I should note that I was able to quickly compute (<1 second) the 3-dimensional analog of the above.