Extremal volume ratios for the sums of normed spaces (Q2773593)

Let \(d(X,Y)\) be the Banach-Mazur distance between \(n\)-dimensional normed spaces \(X\) and \(Y\). Let NEWLINE\[NEWLINE \partial(X,Y) = \inf\{\|T\|_{X\to Y}: |\det T|= 1\} \cdot \inf\{\|T\|_{Y\to X}: |\det T|= 1\} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \text{Vr}(X,Y) = \left(\inf\left\{\frac{\text{vol} B_X}{\text{vol} UB_X}: UB_Y \subset B_X\right\}\right)^{1/n}. NEWLINE\]NEWLINE The aim is to study the distance \(\partial\) and volumetric relation \(\text{Vr}\) for sums of normed spaces. The main results are as follows.NEWLINENEWLINENEWLINETheorem 1. For all normed spaces \(X_1\) and \(X_2\), \(Y_1\) and \(Y_2\) \((\dim X_k = \dim Y_k = n_k\), \(k = 1, 2\)) the following holds: NEWLINE\[NEWLINE \text{Vr}(X_1\oplus X_2, Y_1\oplus Y_2) \leq 2\bigl(\text{Vr}(X_1, Y_1)\bigr)^{\frac{n_1}{n_1 + n_2}} \bigr(\text{Vr}(X_2, Y_2)\bigr)^{\frac{n_2}{n_1 + n_2}}. NEWLINE\]NEWLINE Theorem 2. For all normed spaces \(X\) and \(Y\), \(\dim X = n\), \(\dim Y = k < \delta n\), there exist spaces \(Z\) and \(Z'\) (\(\dim Z = \dim Z' = n - k < \delta n\)) such that NEWLINE\[NEWLINE \text{Vr}(X, Y\oplus Z) \geq c\left(\frac{n}{\ln\ln n}\right)^ {\frac{1 - \delta}{2}},\quad \text{Vr}(Y\oplus Z', X) \geq c\left(\frac{n}{\ln\ln n}\right)^ {\frac{1 - \delta}{2}}, NEWLINE\]NEWLINE where \(c\) is a constant. For \(X = \ell_n^1\) the following inequalities are fulfilled: NEWLINE\[NEWLINE \text{Vr}(\ell_n^1, Y\oplus Z) \geq cn^{\frac{1 - \delta}{2}}, \quad \text{Vr}(Y\oplus Z',\ell_n^{\infty}) \geq cn^{\frac{1 - \delta}{2}}. NEWLINE\]NEWLINE Theorem 3. Let \(0\leq\delta_1,\delta_2 \leq 1\), and \(n_1\leq\delta_1n\), \(n_2\leq\delta_2n\), \(\dim X_0 = n_1\), \(\dim Y_0 = n_2\). Then there exist spaces \(X\) and \(Y\), \(\dim X = n - n_1\), \(\dim Y = n - n_2\) such that NEWLINE\[NEWLINE d(X\oplus X_0, Y\oplus Y_0) \geq \partial (X\oplus X_0, Y\oplus Y_0) \geq c\left(\frac{n}{\ln\ln n}\right)^{1 - \frac{\delta_1+ \delta_2}{2}}. NEWLINE\]NEWLINENEWLINENEWLINENEWLINETheorem 4. If \(\alpha n > n_1 + n_2 \geq n\) for \(\alpha\in (1,2)\) then NEWLINE\[NEWLINE \sqrt n \geq \sup d( X\oplus \ell^2_{n_1},Y\oplus \ell^2_{n_2}) \geq \sqrt{1 - \alpha /2} \sqrt n, NEWLINE\]NEWLINE where \(\dim X = n - n_1\) and \(\dim Y = n - n_2\).NEWLINENEWLINENEWLINETheorem 5. For every \(n \in \mathbb N\) there exists an \(n\)-dimensional space \(X\) such that NEWLINE\[NEWLINE d(X,Y) \geq \partial(X,Y) \geq c\sqrt n NEWLINE\]NEWLINE for every space \(Y\) with \(1\)-unconditional basis and \(d(X,\ell_n^p) \geq \partial(X,\ell_n^p) \geq c\sqrt n\) for \(1 \leq p\leq\infty\).