On area and side lengths of triangles in normed planes
Volume 115 / 2009
Colloquium Mathematicum 115 (2009), 101-112
MSC: Primary 52A21, 52A10; Secondary 52B99.
DOI: 10.4064/cm115-1-9
Abstract
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.
