Positions of convex bodies associated to extremal problems and isotropic measures (Q1826875)

If a convex body minimizes, among all its affine images of the same volume, a certain functional, then this implies in several cases that some measure \(\mu_K\) associated with \(K\) is isotropic (i.e., the matrix \((\int_{\mathbb R^n} x_i x_j\,d\mu_K(x))_{ij}\) with respect to an orthonormal basis is a multiple of the unit matrix). This observation was systematized and applied, in particular, by \textit{A. A. Giannopoulos} and \textit{V. D. Milman} [Isr. J. Math. 117, 29--60 (2000; Zbl 0964.52004)]. In the present paper, the authors study the same phenomenon, with linear instead of affine maps, for functionals arising from the dual Brunn-Minkowski theory. For example, under smoothness assumptions on \(K\), the condition \(\widetilde W_i(K)= \min\{\widetilde W_i(SK): S\in \text{SL}(n)\}\) for the dual quermassintegral \(\widetilde W_i\) and \(i\in (\infty,0)\) (or for \(i\in [n+ 1,\infty)\) and symmetric \(K\)) is equivalent to the fact that the measure given by \(\rho^{n-i}_K\,d\sigma\) (\(\rho_K\) is the radial function) is isotropic on the unit sphere. Also studied are reverse inequalities which are related to some previously studied positions of convex bodies.