Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0407385

Real and discrete holomorphy : Introduction to an algebraic approach

Sylvain Barré·Abdelghani Zeghib
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is then possible to define an "amazing" notion of holomorphic functions on them, and show how rigid it is in some cases.

Explore related subjects

math.MGmath.CVmath.DGmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sylvain Barré, Abdelghani Zeghib. 2007-03-29. Real and discrete holomorphy : Introduction to an algebraic approach. https://arxiv.org/abs/math/0407385

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↗