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Convex meets complex

Yanir A. Rubinstein

Abstract

Convex geometry and complex geometry have long had fascinating interactions. This survey offers a tour of a few.

Convex meets complex

Abstract

Convex geometry and complex geometry have long had fascinating interactions. This survey offers a tour of a few.

Paper Structure

This paper contains 11 sections, 2 theorems, 34 equations.

Table of Contents

  1. Prologue
  2. An early convex--complex result: Study's theorem
  3. Riesz--Thorin and the complex method of interpolation
  4. Berezin--Berndtsson--Lagerberg formalism
  5. Toric geometry as a bridge
  6. Toric manifolds: higher-dimensional surfaces of revolution
  7. The Cauchy problem for the HRMA
  8. Okounkov bodies and Witt Nyström's Chebyshev transform
  9. Complex Legendre duality
  10. A convex-complex approach to the Mahler conjectures
  11. Epilogue

Key Result

Theorem 2.1

Let $f$ be a conformal map on $\mathbb{D}$ whose image is a convex set $C\subset \mathbb{C}$. Let $r\in(0,1)$. The image of $r\mathbb{D}=\{z\in\mathbb{D}\,:\, |z|<r\}$ under $f$ is a convex set.

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • proof : Another proof
  • Theorem 4.1
  • Definition 5.1
  • Definition 8.1
  • Definition 8.2
  • Conjecture 8.3
  • Conjecture 8.4