Hölder continuity for support measures of convex bodies

Hug, Daniel; Schneider, Rolf

doi:10.1007/s00013-014-0719-0

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…We remark that a strengthened form of the continuity assertion (c) of Lemma 4.2, namely local Hölder continuity of the normal cycles of convex bodies with respect to the Hausdorff metric and the dual flat seminorm, is proved in [26].…”

Section: Weakly Continuous Extensionsmentioning

confidence: 95%

Local Tensor Valuations

Hug

Schneider

2014

Geom. Funct. Anal.

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The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continuity.

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Section: Weakly Continuous Extensionsmentioning

confidence: 95%

Local Tensor Valuations

Hug

Schneider

2014

Geom. Funct. Anal.

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“…Similar results hold, for example, if a polyconvex set

K

is approximated by its parallel sets

{K}_{\varepsilon }

; see 50,52 . There are also stability results showing Hölder continuity with exponent at least 1/2 55 If translation invariance is required, then beside the Minkowski functionals among the above mentioned Minkowski tensors precisely

{W}_1&#x0005E;{0,2}

and

{W}_2&#x0005E;{0,2}

are suitable.…”

Section: Using Minkowski Tensors For Describing Microstructuresmentioning

confidence: 64%

Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor

Ernesti

Schneider

Winter

et al. 2022

Math Methods in App Sciences

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For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures.

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“…The fact that the conic support measures are weakly continuous will now be improved, by establishing Hölder continuity with respect to a metric which metrizes the weak convergence. This is in analogy to the case of support measures of convex bodies, which was treated in [18].…”

Section: Hölder Continuity Of the Conic Support Measuresmentioning

confidence: 99%