Marginals of Geometric Inequalities

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Some functional forms of Blaschke–Santaló inequality (2006, background): “y for symmetric sets and some class of rotation invariant measures. This inequality is known for the Lebesgue measure and the Gaussian measure (see [7]); and also for a special class of measures (see [12]). It was asked in [7] whether it holds for any symmetric log-concave measure. We also give here”
Uniform almost sub-Gaussian estimates for linear functionals on convex sets (2007, methodology): “It follows directly from the Brunn-Minkowski inequality (1), see [ 19 ].”
Super currents and tropical geometry (2010, background): “Moreover, the setting presented here seems to share similarities with that of [7].”
Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (2011, methodology): “However, we may take the derivative at s = +∞ as in [10] (the details are given in the proof below), and doing so, we obtain the inequality of Theorem 1.”