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Isoperimetric inequalities for \(L_{p}\) geominimal surface area. (English) Zbl 1235.52019
Let \(K,L\) be two convex bodies containing the origin as an interior point, which is denoted by \(K,L\in{\mathcal{K}}^n_0\). Then, for any \(p\geq 1\), the \(p\)-sum (or Firey combination) of \(K\) and \(L\) is defined via their support functions by \(h(K+_pL,u)^p=h(K,u)^p+h(L,u)^p\) for all \(u\in{\mathbb{R}}^n\backslash\{0\}\). This operation was introduced and studied by W. J. Firey in [Math. Scand. 10, 17–24 (1962; Zbl 0188.27303)]. Later, E. Lutwak [J. Differ. Geom. 38, No. 1, 131–150 (1993; Zbl 0788.52007)] defined the \(L_p\)-mixed volume by \[ V_p(K,L)=\frac{p}{n}\lim_{\varepsilon\rightarrow 0}\frac{V(K+_p\varepsilon L)-V(K)}{\varepsilon}, \] (here \(V\) denotes the volume functional), and studied \(p\)-sums of convex bodies systematically, developing a theory nowadays known as Brunn-Minkowski-Firey theory. Related to this concept, the \(L_p\)-geominimal surface area can be defined as \[ G_p(K)=V(B_n)^{-p/n}\inf_{L\in\mathcal{K}^n_0}\left\{nV_p(K,L)V(L^*)^{p/n}\right\}, \] where \(B\) denotes the Euclidean unit ball and \(L^*\) is the polar body of \(L\).
In this paper the authors establish several inequalities for \(G_p(K)\), namely, a cyclic inequality for \(G_p(K)\), a Blaschke-Santaló type inequality and an affine isoperimetric type inequality. More precisely, they show that for \(K\in\mathcal{K}^n_0\) and \(1\leq p<q<r\) it holds that \(G_q(K)^{r-p}\leq G_p(K)^{r-q}G_r(K)^{q-p}\), and moreover, if \(K\) is in addition symmetric with respect to the origin, then \[ G_p(K)G_p(K^*)\leq n^2V(B_n)^2\quad\text{ and }\quad G_p(K)^n\leq n^nV(B_n)^{n-p}V(\Pi_pK)^p, \] with equality in the last two inequalities if and only if \(K\) is an ellipsoid; here \(\Pi_pK\) represents the so called \(L_p\)-projection body of \(K\).

MSC:
52A40 Inequalities and extremum problems involving convexity in convex geometry
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