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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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