The Mahler Conjecture in Two Dimensions via the Probabilistic Method
| Autor: | Matthew Tointon |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Conjecture Mathematics::Number Theory General Mathematics 010102 general mathematics Metric Geometry (math.MG) 01 natural sciences Measure (mathematics) Mahler volume Probabilistic method Mathematics - Metric Geometry 0103 physical sciences Intuition (Bergson) FOS: Mathematics Mathematics - Combinatorics Convex body Combinatorics (math.CO) 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | The American Mathematical Monthly. 125:820-828 |
| ISSN: | 1930-0972 0002-9890 |
| Popis: | The "Mahler volume" is, intuitively speaking, a measure of how "round" a centrally symmetric convex body is. In one direction this intuition is given weight by a result of Santalo, who in the 1940s showed that the Mahler volume is maximized, in a given dimension, by the unit sphere and its linear images, and only these. A counterpart to this result in the opposite direction is proposed by a conjecture, formulated by Kurt Mahler in the 1930s and still open in dimensions 4 and greater, asserting that the Mahler volume should be minimized by a cuboid. In this article we present a seemingly new proof of the 2-dimensional case of this conjecture via the probabilistic method. The central idea is to show that either deleting a random pair of edges from a centrally symmetric convex polygon, or deleting a random pair of vertices, reduces the Mahler volume with positive probability. Comment: 9 pages, 6 figures. Minor corrections from original version. To appear in Amer. Math. Monthly |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|