Consider 4 4 vectors

$$

P_ {0} = (E_ {0}, 0,0, sqrt {E_ {0} ^ {2} -m_ {0} ^ {2}}), quad P_ {i} = (E_ {i} , p_ {i} s ( theta_ {i}) c ( phi_ {i}), p_ {i} s ( phi_ {i}) s ( theta_ {i}), p_ {i} c ( theta_ {i})),

$$

with $ c equiv cos, s equiv sin $, $ p_ {i} equiv sqrt {E_ {i} ^ {2} -m_ {i} ^ {2}} $ and dot products

$$

P_ {i} cdot P_ {j} equiv P_ {i} ^ {0} P_ {j} ^ {0} – sum_ {k = 1} ^ {3} P_ {i} ^ {k} P_ { j} ^ {k}

$$

$ m_ {0-3}, E_ {0} $ play the role of real parameters, with $ E_ {0}> m_ {0}> m_ {1} + m_ {2} + m_ {3} $ and $ E_ {i} geqslant m_ {i} $, while $ E_ {i}, theta_ {i}, phi_ {i} $ are variables.

The implicit region of the definition of $ E_ {i}, theta_ {i}, phi_ {i} $ is given by

$$

tag 1 P_ {3} = P_ {0} -P_ {1} -P_ {2},

$$

$$

tag 2 s_ {12, text {min}} (s_ {23}) <s_ {12} <s_ {12, text {max}} (s_ {23}), quad s_ {23, text {min}} <s_ {23} <s_ {23, text {max}},

$$

or $ s_ {ij} = m_ {i} ^ {2} + m_ {j} ^ {2} + 2P_ {i} cdot P_ {j} $, and

$$

tag 3 s_ {12, text {min} / text {max}} = m_ {1} ^ {2} + m_ {2} ^ {2} – frac {1} {2s_ {23}} bigg (s_ {23} -m_ {0} ^ {2} + m_ {1} ^ {2}) (s_ {23} -m_ {2} ^ {2} -m_ {3} ^ {2}) pm \ pm sqrt { lambda (s_ {23}, m_ {0} ^ {2}, m_ {1} ^ {2}) lambda (s_ {23}, m_ {2} ^ {2} , m_ {3} ^ {2})} bigg),

$$

$$

tag 4 s_ {23, text {min}} = (m_ {2} + m_ {3}) ^ {2}, quad s_ {23, text {max}} = (m_ {0} -m_ {1}) ^ {2}, quad lambda (a, b, c) = (abc) ^ {2} -4bc

$$

I need to integrate a function $ f (E_ {i}, theta_ {i}, phi_ {i}) $ on the domain of definition $ (1) – (4) $ of the variables mentioned. Is it possible to derive the domain of definition in Mathematica, at least implicitly, in order to perform integration? There are so many variables …