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Upper envelopes of inner premeasures

Heinz König (2000)

Annales de l'institut Fourier

The paper resumes one of the themes initiated in the final sections of the celebrated “Theory of Capacities” of Choquet 1953-54. It aims at comprehensive versions in the spirit of the author’s recent work in measure and integration.

Upper Estimate of Concentration and Thin Dimensions of Measures

H. Gacki, A. Lasota, J. Myjak (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

Upper estimates on self-perimeters of unit circles for gauges

Horst Martini, Anatoliy Shcherba (2016)

Colloquium Mathematicae

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.

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