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1
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On the stability of the unit circle with minimal self-perimeter in normed planes

100%
Martini H., Shcherba A.
Colloquium Mathematicum
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2013
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tom 131
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nr 1
69-87
EN
We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.
2
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On area and side lengths of triangles in normed planes

100%
Averkov G., Martini H.
Colloquium Mathematicum
|
2009
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tom 115
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nr 1
101-112
EN
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.
3
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Upper estimates on self-perimeters of unit circles for gauges

100%
Martini H., Shcherba A.
Colloquium Mathematicum
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2016
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tom 142
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nr 2
179-210
EN
Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.
4
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On unit balls and isoperimetrices in normed spaces

100%
Martini H., Mustafaev Z.
Colloquium Mathematicum
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2012
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tom 127
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nr 1
133-142
EN
The purpose of this paper is to continue the investigations on the homothety of unit balls and isoperimetrices in higher-dimensional Minkowski spaces for the Holmes-Thompson measure and the Busemann measure. Moreover, we show a strong relation between affine isoperimetric inequalities and Minkowski geometry by proving some new related inequalities.
5
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Minkowskian rhombi and squares inscribed in convex Jordan curves

100%
Martini H., Wu S.
Colloquium Mathematicum
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2010
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tom 120
|
nr 2
249-261
EN
We show that any convex Jordan curve in a normed plane admits an inscribed Minkowskian square. In addition we prove that no two different Minkowskian rhombi with the same direction of one diagonal can be inscribed in the same strictly convex Jordan curve.
6
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Estimates on inner and outer radii of unit balls in normed spaces

100%
Martini H., Mustafaev Z.
Colloquium Mathematicum
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2011
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tom 123
|
nr 2
211-217
EN
The purpose of this paper is to continue the investigations on extremal values for inner and outer radii of the unit ball of a finite-dimensional real Banach space for the Holmes-Thompson and Busemann measures. Furthermore, we give a related new characterization of ellipsoids in $ℝ^{d}$ via codimensional cross-section measures.
7
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Rotation indices related to Poncelet’s closure theorem

81%
Cieślak W., Martini H., Mozgawa W.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2014
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tom 68
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nr 2
EN
Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet’s closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet’s closure theorem together with Cr, with n- gons for any n > k.
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