Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity
Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry which associates to points polar hyperplanes, and more generally $k$-dimensional convex bodies to polar $(n-1-k)$-dimensional convex bodies. It is well-known that the Legendre-Fenchel transformation of functions can be interpreted from the polarity viewpoint of their graphs using an extra dimension. In this paper, we first show that generic polarities induced by quadratic polarity functionals can be expressed either as deformed Legendre polarity or as the Legendre polarity of deformed convex bodies, and be efficiently manipulated using linear algebra on $(n+2)\times (n+2)$ matrices operating on homogeneous...
| Authors | Frank Nielsen, Basile Plus-Gourdon, Mahito Sugiyama |
|---|---|
| Primary category | cs.CG |
| Published | 5 March 2026 |
| Last updated at source | 5 March 2026 |
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