Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/ √ d) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of nondegeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d 2). We conjecture that both bounds are optimal for triangulations in Euclidean space.

We quantise Whitney’s construction to prove the existence of a triangulation for any $$C^2$$ C 2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

Today's biometric authentication systems are still struggling with replay attacks and irrevocable stolen credentials. This paper introduces a biometric protocol that addresses such vulnerabilities. The approach prevents identity theft by being based on memory creation biometrics. It takes inspiration from two different authentication methods, eye biometrics and challenge systems, as well as a novel biometric feature: the pupil memory effect. The approach can be adjusted for arbitrary levels of security, and credentials can be revoked at any point with no loss to the user. The paper includes an analysis of its security and performance, and shows how it could be deployed and improved.

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