Olof Hanner (7 December 1922 in

Stockholm Stockholm () is the Capital city, capital and List of urban areas in Sweden by population, largest city of Sweden as well as the List of urban areas in the Nordic countries, largest urban area in Scandinavia. Approximately 980,000 people liv ...

– 19 September 2015 in

Gothenburg Gothenburg (; abbreviated Gbg; sv, Göteborg ) is the second-largest city in Sweden, fifth-largest in the Nordic countries, and capital of the Västra Götaland County. It is situated by the Kattegat, on the west coast of Sweden, and has ...

) was a

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

mathematician.

Education and career

Hanner earned his Ph.D. from

Stockholm University Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, so ...

in 1952. He was a professor at the

University of Gothenburg The University of Gothenburg ( sv, Göteborgs universitet) is a university in Sweden's second largest city, Gothenburg. Founded in 1891, the university is the third-oldest of the current Swedish universities and with 37,000 students and 6000 st ...

from 1963 to 1989.

Contributions

In a 1956 paper, Hanner introduced the

Hanner polytope In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.. Construction The Hanner polytopes are construct ...

s and the Hanner spaces having these polytopes as their metric balls. Hanner was interested in a

Helly property In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non ...

of these shapes, later used to characterize them by : unlike other

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s, it is not possible to find three

translated Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

copies of a Hanner polytope that intersect pairwise but do not have a point of common intersection. Subsequently, the Hanner polytopes formed a class of important examples for the

Mahler conjecture In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is ...

and for Kalai's 3^''d'' conjecture. In another paper from the same year, Hanner proved a set of inequalities related to the uniform convexity of ''L''^''p'' spaces, now known as

Hanner's inequalities In mathematics, Hanner's inequalities are results in the theory of ''L'p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L'p'' spaces for ''p'' ∈ (1,& ...

. Other contributions of Hanner include (with

Hans Rådström Hans Vilhem Rådström (1919–1970) was a Swedish mathematician who worked on complex analysis, continuous groups, convex sets, set-valued analysis, and game theory. From 1952, he was ''lektor'' (assistant professor) at Stockholm Univers ...

) improving

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theor ...

's version of

Carathéodory's lemma In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used this criterion to prove ...

, contributing to ''

The Official Encyclopedia of Bridge ''The Official Encyclopedia of Bridge'' (OEB) presents comprehensive information on the card game contract bridge with limited information on related games and on playing cards. It is "official" in reference to the American Contract Bridge League ...

'', and doing early work on

combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

and the

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

of the board game Go. One of the many proofs of the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

based on the

Pythagorean tiling A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are ...

is sometimes called "Olof Hanner's Jigsaw Puzzle".Olof Hanner's Jigsaw Puzzle

Cut-the-Knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

, retrieved 2013-05-17.

Selected publications

*. *. *. *. *. *. *

References

External links

Olof Hanner
at

WorldCat WorldCat is a union catalog that itemizes the collections of tens of thousands of institutions (mostly libraries), in many countries, that are current or past members of the OCLC global cooperative. It is operated by OCLC, Inc. Many of the OCL ...

{{DEFAULTSORT:Hanner, Olof 1922 births 2015 deaths Swedish mathematicians Stockholm University alumni University of Gothenburg faculty