Geodesic
On area and side lengths of triangles in normed planes
Gennadiy Averkov 1   ; Horst Martini 2
1 Faculty of Mathematics University of Magdeburg 39106 Magdeburg, Germany
2 Faculty of Mathematics University of Technology 09107 Chemnitz, Germany
Colloquium Mathematicum, Tome 115 (2009) no. 1, pp. 101-112 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $\mathcal M^d$ be a $d$-dimensional normed space with norm $\|\,\cdot\, \|$ and let $B$ be the unit ball in $\mathcal M^d.$ Let us fix a Lebesgue measure $V_B$ in $\mathcal M^d$ with $V_B(B)=1.$ This measure will play the role of the volume in $\mathcal M^d$. We consider an arbitrary simplex $T$ in $\mathcal M^d$ with prescribed edge lengths. For the case $d=2$, sharp upper and lower bounds of $V_B(T)$ are determined. For $d\ge 3$ it is noticed that the tight lower bound of $V_B(T)$ is zero.
DOI : 10.4064/cm115-1-9
Keywords: mathcal d dimensional normed space norm cdot unit ball mathcal fix lebesgue measure mathcal measure play role volume mathcal consider arbitrary simplex mathcal prescribed edge lengths sharp upper lower bounds determined noticed tight lower bound zero

Gennadiy Averkov  1   ; Horst Martini  2

1 Faculty of Mathematics University of Magdeburg 39106 Magdeburg, Germany
2 Faculty of Mathematics University of Technology 09107 Chemnitz, Germany 
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Gennadiy Averkov; Horst Martini. On area and side lengths  of triangles in normed planes. Colloquium Mathematicum, Tome 115 (2009) no. 1, pp. 101-112. doi: 10.4064/cm115-1-9

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