@article{6515,
abstract = {We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.},
author = {Dyer, Ramsay and Vegter, Gert and Wintraecken, Mathijs},
issn = {1920-180X},
journal = {Journal of Computational Geometry },
number = {1},
pages = {223–256},
title = {{Simplices modelled on spaces of constant curvature}},
doi = {10.20382/jocg.v10i1a9},
volume = {10},
year = {2019},
}
@inproceedings{6628,
abstract = {Fejes Tóth [5] and Schneider [9] studied approximations of smooth convex hypersurfaces in Euclidean space by piecewise flat triangular meshes with a given number of vertices on the hypersurface that are optimal with respect to Hausdorff distance. They proved that this Hausdorff distance decreases inversely proportional with m 2/(d−1), where m is the number of vertices and d is the dimension of Euclidean space. Moreover the pro-portionality constant can be expressed in terms of the Gaussian curvature, an intrinsic quantity. In this short note, we prove the extrinsic nature of this constant for manifolds of sufficiently high codimension. We do so by constructing an family of isometric embeddings of the flat torus in Euclidean space.},
author = {Vegter, Gert and Wintraecken, Mathijs},
booktitle = {The 31st Canadian Conference in Computational Geometry},
location = {Edmonton, Canada},
title = {{The extrinsic nature of the Hausdorff distance of optimal triangulations of manifolds}},
year = {2019},
}
@article{6050,
abstract = {We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. },
author = {Akopyan, Arseniy and Fedorov, Roman},
journal = {Proceedings of the American Mathematical Society},
pages = {91--102},
publisher = {AMS},
title = {{Two circles and only a straightedge}},
doi = {10.1090/proc/14240},
volume = {147},
year = {2019},
}
@article{6608,
abstract = {We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram.},
author = {Edelsbrunner, Herbert and Ölsböck, Katharina},
journal = {Computer Aided Geometric Design},
pages = {1--15},
publisher = {Elsevier},
title = {{Holes and dependences in an ordered complex}},
doi = {10.1016/j.cagd.2019.06.003},
volume = {73},
year = {2019},
}
@article{6634,
abstract = {In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.},
author = {Akopyan, Arseniy and Hubard, Alfredo and Karasev, Roman},
journal = {Topological Methods in Nonlinear Analysis},
number = {2},
pages = {457--490},
publisher = {Akademicka Platforma Czasopism},
title = {{Lower and upper bounds for the waists of different spaces}},
doi = {10.12775/TMNA.2019.008},
volume = {53},
year = {2019},
}