For each pair of numbers \(m,n\in {{\mathbb {N}}}\) with \(m>n\), we consider the norm on \({{\mathbb {R}}}^3\) given by \(\Vert (a,b,c)\Vert _{m,n}=\sup \{|ax^m+bx^{m-n}y^n+cy^m|:x,y\in [-1,1]\}\) for every \((a,b,c)\in {{\mathbb {R}}}^3\). We investigate some geometrical properties of these norms. We provide an explicit formula for \(\Vert \cdot \Vert _{m,n}\), a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres for certain values of m and n.

中文翻译：

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$${\mathbb {R}}^2$$ R 2 上齐次三项式的空间几何

对于每对数\(m,n\in {{\mathbb {N}}}\)和\(m>n\)，我们考虑\({{\mathbb {R}}}}^3上的范数\)由\(\Vert (a,b,c)\Vert _{m,n}=\sup \{|ax^m+bx^{mn}y^n+cy^m|:x,y \in [-1,1]\}\)对于每个\((a,b,c)\in {{\mathbb {R}}}^3\)。我们研究了这些范数的一些几何特性。我们为\(\Vert \cdot \Vert _{m,n}\)提供了一个明确的公式，对相应单位球的极值点的完整描述和参数化以及它们对于某些m值的单位球体的绘图和n。