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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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