Polar Varieties in Cayley-Klein Spaces

Fahimeh Heidari; Bijan Honari

Home/Archive/arXiv

arXiv · 2201.10330

Polar Varieties in Cayley-Klein Spaces

Fahimeh Heidari·Bijan Honari

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce the notion of a total polar for an arbitrary subspace of a Cayley-Klein space in an analytical framework. We show that the set of all total polars of a subspace is a Schubert variety. The notion of total polar gives a definition for a subspace to be tangent to the absolute figure of the space. By specifying tangent lines, tangent cones and then spheres are defined. This definition of the sphere does not depend on the metric of the space. It is proved that every reflection of a Cayley- Klein space, defined by two subspaces which are total polar to each other, is a motion of the space. On the other hand, each motion in a Cayley-Klein space of dimension n is a product of at most n+1 reflections in point-hyperplane pairs.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fahimeh Heidari, Bijan Honari. 2022-01-25. Polar Varieties in Cayley-Klein Spaces. https://arxiv.org/abs/2201.10330

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Godbersen's conjecture and the $L_p$-Rogers-Shephard inequality

We prove that the mixed volume of a convex body of fixed positive volume with its reflection about the origin is maximized by simplices. This confirms a conjecture of C. Godbersen from 1938 and refines the classical Rogers-Shephard inequality. We also prove that simplices are the only extremizers among convex polytopes. Finally, we use this inequality to prove an $L_p$-version of the Rogers-Shephard inequality for convex bodies containing the origin and show that, for any $p\in(1,\infty]$, the only extremizers are simplices with a vertex at the origin.

math.MG↗

arXiv2026

Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is sharp to the linear order in $\varepsilon$, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor $\sqrt{n s(K)}$. Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the $L_p$-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the $L_p$-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.

math.MG↗

arXiv2026

Busemann G-spaces with convex balls

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii--Halverson--Repovš.

math.MG↗