On the Goncharov depth conjecture and a formula for volumes of orthoschemes

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms depth at most half the weight. give an explicit formula for this presentation, involving summation over trees correspond to decompositions polygon into quadrangles. Our second result is volume hyperbolic orthoschemes, generalizing Lobachevsky in dimension three arbitrary dimension. show surprising relation between two results, comes from fact orthoschemes are parametrized by configurations points on P 1 \mathbb {P}^1 . In particular, we derive both formulas their common generalization.