Integrals over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices $(1,0)$, $(0,1)$, $(1,1)$. We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as linear combinations of multiple zeta values, and as polynomials in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant, and study integrals, one of which is the iterated Chen (Drinfeld–Kontsevich) integral, over some polytopes that are higher-dimensional analogs of $T$. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.