On unit balls and isoperimetrices in normed spaces (Q2893058)

Let \((\mathbb{R}^d, \|\cdot\|)\) be a finite dimensional real Banach space with unit ball \(B\). Let \(\widehat{I}_B^{\mathrm{HT}}\) denote the corresponding isoperimetrix for the Holmes-Thompson measure and \(\widehat{I}_B^{\mathrm{Bus}}\) the isoperimetrix for the Busemann measure (for the respective definitions see, e.g., Chapter 5 of [\textit{A. C. Thompson}, Minkowski geometry. Cambridge: Cambridge University Press (1996; Zbl 0868.52001)]). In the present paper, some isoperimetric inequalities are proved and relations between the unit ball and the respective isoperimetrices are investigated. Two of the results say the following: if \(B\) is a centered body of cylindrical type in \(\mathbb{R}^d\) with \(d\geq 3\), then \(B\) and \(\widehat{I}_B^{\mathrm{HT}}\) cannot be homothetic; if \(B\) is a centered convex body of the type of a double cone in \(\mathbb{R}^d\) with \(d\geq 3\), then \(B\) and \(\widehat{I}_B^{\mathrm{Bus}}\) cannot be homothetic either. Also, the authors prove an estimate related to Petty's conjectured projection inequality.