Extreme properties of quermassintegrals of convex bodies
The authors prove inequalities involving quermassintegrals of projections of convex bodies. One main result says (after correcting a misprint) that, for unit vectors \(u_1,\dots,u_m\) in \(\mathbb{R}^n\) and real numbers \(c_1,\dots, c_m\) satisfying \(\sum^m_{i=1} c_iu_i\otimes u_i=I_n\) (identity), one has \[ W_j(K)^{n-j-1} \leq\omega_n^{-j/n} \prod^m_{i=1} \bigl\{W_j(K^{u_i}) \bigr \}^{c_i (n-j)/n} \] for every convex body \(K\) with interior points in \(\mathbb{R}^n\) and for \(0\leq j\leq n-1\). Here \(W_j\) is the \(j\)th quermassintegral and \(K^u\) denotes the image of \(K\) under orthogonal projection to \(u^\perp\) \((\omega_n\) is the volume of the \(n\)-dimensional unit ball). For \(j=0\), the result reduces to a theorem of \textit{K. Ball} [Trans. Am. Math. Soc. 327, No. 2, 891-901 (1991; Zbl 0746.52007)]. The proof follows the one by Ball, and in addition uses known inequalities for quermassintegrals. The paper contains further results on quermassintegrals of projections. Reviewer's remark. The other main result, Theorem 2 (also marred by misprints: the assumptions should be \(K\in\mathbb{K}^n\), \(C\in\mathbb{K}^n_s\); cf. Corollary 1), is contained in earlier results by \textit{G. D. Chakerian} and \textit{E. Lutwak} [Trans. Am. Math. Soc. 349, No. 5, 1811-1820 (1997; Zbl 0873.52010)].
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