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The \(L_{p}\)-Mixed Geominimal Surface Areas

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Abstract

In the paper, our main aim is to introduce a new concept and call it the \(L_p\)- mixed geominimal surface area \(G_{p}(K_{1},\,\dots\,,K_{n})\) of \(n\) convex bodies \(K_{1},\,\dots\,,K_{n}\), which obeys the classical basic properties. The new affine geometric quantity in a special case yields Petty’s geominimal surface area \(G(K)\) of a convex body \(K\), Lutwak’s \(p\)-geominimal surface area \(G_{p}(K)\) of \(K\), and the newly established mixed geominimal surface area \(G(K_{1},\,\dots\,,K_{n})\) of \(n\) convex bodies \(K_{1},\,\dots\,,K_{n}\). We establish some \(L_{p}\)-mixed geominimal surface area inequalities for the \(L_p\)-mixed geominimal surface area, whose some special cases are Petty’s geominimal surface area inequality, Lutwak’s \(p\)-geominimal surface area inequality, and some new mixed geominimal surface area inequalities.

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  1. China Jiliang University, Hangzhou, 310018, P. R. China

    C. J. Zhao

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  1. C. J. Zhao
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Correspondence to C. J. Zhao.

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Zhao, C.J. The \(L_{p}\)-Mixed Geominimal Surface Areas. Math Notes 112, 1044–1058 (2022). https://doi.org/10.1134/S0001434622110360

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