We study properties of the volume of projections of the $n$-dimensional cross-polytope ${♢}^n=\{x\in {\mathbb{R}}^n\mid \left|x_1\right|+\cdots +\left|x_n\right|⩽1\}$. We prove that the projection of ${♢}^n$ onto a $k$-dimensional coordinate subspace has the maximum possible volume for $k=2$ and for $k=3.$ We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of ${♢}^n$ onto a $k$-dimensional subspace for any $n>k\ge 2.$

中文翻译：

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关于交叉多面体的投影量

我们研究了投影的体积特性 $ñ$维交叉多态性 ${♢}^{ñ}=\{X\in {\mathbb{[R}}^{ñ}\mid \left|X_{\mathrm{1个}}\right|+\cdots +\left|X_{ñ}\right|⩽\mathrm{1个}\}$。我们证明${♢}^{ñ}$ 到一个 $ķ$维坐标子空间具有最大可能的体积 $ķ=2$ 和为 $ķ=3。$我们获得了这种投影到二维平面上的体积的确切下界。此外，我们表明存在局部最大值，对于投影的体积，该最大值不是全局最大值${♢}^{ñ}$ 到一个 $ķ$维子空间 $ñ>ķ\ge 2。$