## Inequalities for mixed $$p$$-affine surface area.(English)Zbl 1192.52013

Let $$\mathcal{C}^2_+$$ be the class of all convex bodies in $$\mathbb{R}^n$$ with $$\mathcal{C}^2$$ boundary, everywhere strictly positive Gaussian curvature and whose centroid is at the origin.
In 1996 Lutwak defined for $$p\in [1, +\infty[$$ the notion of mixed $$p$$-affine surface area $$as_p(K_1,\dots ,K_n)$$ of $$n$$ convex bodies $$K_i\in \mathcal{C}^2_+$$ by
$as_p(K_1,\dots ,K_n)=\int_{S^{n-1}}\left[h_{K_1}(u)^{1-p}f_{K_1}(u)\dots h_{K_n}(u)^{1-p}f_{K_n}(u)\right]^{\frac{1}{n+p}}d\sigma (u).$
Here $$S^{n-1}$$ is the boundary of the euclidean unit ball $$B^n_2$$ in $$\mathbb{R}^n$$. $$\sigma$$ is the usual surface area measure on $$S^{n-1}$$, $$h_K(u)$$ is the support function of the convex body $$K$$ at $$u\in S^{n-1}$$ and $$f_K(u)$$ is the curvature function of $$K$$ at $$u$$, i.e., the reciprocal of the Gauss curvature $$\kappa_K(x)$$ at the point $$x\in \partial K$$ with $$u$$ being its outer normal vector.
In this paper the authors extend the above definition to all $$p\in [-\infty, \infty]$$.
They show that mixed $$p$$-affine surface areas are affine invariants for all $$p$$.
They prove new Aleksandrov-Fenchel type inequalities and new isoperimetric inequalities for mixed $$p$$-affine surface areas, and show the monotonicity behaviour of the quotients
$\left(\frac{as_p(K_1,\dots ,K_n)}{as_{\infty}(K_1,\dots ,K_n)}\right)^{n+p}$
and
$\left(\frac{as_p(K_1,\dots ,K_n)}{as_0(K_1,\dots ,K_n)}\right)^{\frac{n+p}{p}}.$
They also prove Blaschke-Santaló type inequalities for mixed $$p$$-affine surface areas.
Furthermore, they extend for all $$p$$ the notion of $$i$$th mixed $$p$$-affine surface area and prove the corresponding inequalities.
They introduce a new class of bodies – the illumination surface bodies – and establish some of their properties; these bodies are not necessarily convex. Using these bodies they obtain interesting geometric interpretations of $$L_p$$-affine surface areas, mixed $$p$$-affine surface areas and other functionals.

### MSC:

 52A39 Mixed volumes and related topics in convex geometry 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A40 Inequalities and extremum problems involving convexity in convex geometry 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53A15 Affine differential geometry
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