Applications of various inequalities to Minkowski geometry
A finite-dimensional normed linear space \(X\) with a unit ball \(B\) is equipped with an auxiliary Euclidean structure which allows to identify \(X\) with some \(\mathbb{R}^ d\) and to normalize the Haar measure in each \(k\)- dimensional subspace of \(X\). In each subspace the Minkowski content \(\mu\) is a multiple of the appropriate Haar measure \(\lambda\). Here the author deals with the most interesting case \(k=d-1\) of hyperplanes of \(X\). Then \(\mu=\sigma(f)\lambda\), where \(f\) is the unit normal to \(H\), that is, \(f\) is a linear functional in the dual space \(X^*\) of \(X\) of Euclidean length 1 whose null space \(f^ \perp\) is \(H\). The author gives lower bounds for the self-surface-area of a ball in \(\mathbb{R}^ d\). More precisely, let \(\mu^{(1)}_ B\) and \(\mu^{(2)}_ B\) denote the area measures which arise from \(\sigma_ 1(f)= {\varepsilon_{d-1}\over \lambda(B\cap f^ \perp)}\) and \(\sigma_ 2(f)= {\lambda((B\cap f^ \perp)^ 0)\over \varepsilon_{d-1}}\) respectively, where \(\varepsilon_ d\) denotes the \(d\)-dimensional volume of a \(d\)- dimensional Euclidean unit ball and \(C^ 0\subseteq X^*\) is the dual of \(C\subseteq X\). Then \(\mu^{(1)}_ B(\partial B)\geq (d\varepsilon_ d)\Bigl({m_ d\over \varepsilon^ 2_ d}\Bigr)^{1/d}\) and \(\mu^{(2)}_ B(\partial B)\geq\Bigl({d\over\varepsilon_ d}\Bigr)\Bigl({4^ d\over d!}\Bigr)^{1/d}(m_ d)^{(d-1)/d}\), where \(m_ d\) is the minimum of \(\lambda(K)\lambda(K^ 0)\) taken over all centrally symmetric convex bodies \(K\) in \(\mathbb{R}^ d\). These lower bounds follow from the Minkowski inequality for mixed volumes, the Petty projection inequality, the Busemann intersection inequality and the Mahler-Reisner inequality. For 2-dimensional Minkowski spaces the lower bounds obtained are surprisingly close to the actual bound 6. For \(d=3\), and assuming Mahler's conjecture (i.e. \(m_ d={4^ d\over d }\)), the given lower bounds improve previous results. The exact values of \(\mu^{(1)}_ B(\partial B)\) and \(\mu^{(2)}_ B(\partial B)\) are calculated for three balls in \(\mathbb{R}^ d\), an ellipsoid, a cube and a zonotope. This yields several conjectures for which balls \(\mu^{(i)}_ B(\partial B)\) is maximal or minimal. Using the same inequalities the author further obtains a characterization of ellipsoids as those balls that contain their isoperimetrix.
scientific article; zbMATH DE number 205535
scientific article