# Simplicial Vertex-Normal Duality with Applications to Well-Centered Simplices

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## Abstract

We study the relation between the set of *n* + 1 vertices of an *n*-simplex *S* having \(\mathbb {S}^{n-1}\) as circumsphere, and the set of *n* + 1 unit outward normals to the facets of *S*. These normals can in turn be interpreted as the vertices of another simplex \(\hat {S}\) that has \(\mathbb {S}^{n-1}\) as circumsphere. We consider the iterative application of the map that takes the simplex *S* to \(\hat {S}\), study its convergence properties, and in particular investigate its fixed points. We will also prove some statements about well-centered simplices in the above context.

## Notes

### Acknowledgements

Michal Křížek was supported by grant no. 18-09628S of the Grant Agency of the Czech Republic.

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