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.
@article{10_4064_cm115_1_9,
author = {Gennadiy Averkov and Horst Martini},
title = {On area and side lengths of triangles in normed planes},
journal = {Colloquium Mathematicum},
pages = {101--112},
year = {2009},
volume = {115},
number = {1},
doi = {10.4064/cm115-1-9},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm115-1-9/}
}
TY - JOUR
AU - Gennadiy Averkov
AU - Horst Martini
TI - On area and side lengths of triangles in normed planes
JO - Colloquium Mathematicum
PY - 2009
SP - 101
EP - 112
VL - 115
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4064/cm115-1-9/
DO - 10.4064/cm115-1-9
LA - en
ID - 10_4064_cm115_1_9
ER -
%0 Journal Article
%A Gennadiy Averkov
%A Horst Martini
%T On area and side lengths of triangles in normed planes
%J Colloquium Mathematicum
%D 2009
%P 101-112
%V 115
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4064/cm115-1-9/
%R 10.4064/cm115-1-9
%G en
%F 10_4064_cm115_1_9
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
