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We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,M_N$ generated by an N-tuple of continuous monotone functions $f₁,...,f_N:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N\ni b\mapsto (M₁\left(b\right),...,M_N\left(b\right))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,M_N$ taken on b. We effectively...

We present both necessary and sufficient conditions for a convex closed shape such that for every convex function the average integral over the shape does not exceed the average integral over its boundary. It is proved that this inequality holds for n-dimensional parallelotopes, n-dimensional balls, and convex polytopes having the inscribed sphere (tangent to all its facets) with the centre in the centre of mass of its boundary.