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Circumradius versus side lengths of triangles in linear normed spaces

100%

Averkov G.

Colloquium Mathematicum

2007

tom 107

nr 2

273-285

Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by $R_B(T)$ the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science $R_B(T)$ is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein). Using methods of linear algebra and convex geometry we find the lower and upper bound of $R_B(T)$ for the case when B is an arbitrary planar convex body centered at the origin and T ⊆ ℳ ²(B) is an arbitrary triangle with given Minkowskian side lengths a₁, a₂, a₃. We also obtain some further results from the geometry of triangles in Minkowski planes, which are either corollaries of the main result or statements needed in the proof of the main result.

On area and side lengths of triangles in normed planes

63%

Averkov G., Martini H.

Colloquium Mathematicum

2009

tom 115

nr 1

101-112

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

Ograniczanie wyników

2 Colloquium Mathematicum

2 Averkov G.

1 Martini H.

1 2009

1 2007

Wyniki wyszukiwania

Circumradius versus side lengths of triangles in linear normed spaces

On area and side lengths of triangles in normed planes