# Olof Hanner
**Olof Hanner** (1922–2015) was a Swedish mathematician, professor at the [University of Gothenburg](https://www.edgepedia.org/university-of-gothenburg) from 1963 to 1989, whose name is attached to two distinct results: the Hanner inequality, which gives the best possible modulus of convexity (A measure of how strongly a space's unit ball curves) for the spaces Lp and lp, and the Hanner polytopes, the conjectured minimizers of the symmetric Mahler volume-product problem in convex geometry.[1](https://www.ne.se/uppslagsverk/encyklopedi/l%C3%A5ng/olof-hanner) • [2](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full)
| Key fact | Detail |
|---|---|
| Life | 1922–2015; professor at the University of Gothenburg 1963–89[1](https://www.ne.se/uppslagsverk/encyklopedi/l%C3%A5ng/olof-hanner) |
| Doctorate | Ph.D., Stockholm University, 1952; dissertation *Retraction and extension of mappings*; advisor Fritz Carlson[3](https://mathgenealogy.org/id.php?id=20636) |
| Hanner inequality | 1956 paper in *Arkiv för Matematik* 3(3):239–244 giving the best possible modulus of convexity for Lp and lp, p > 1[2](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full) |
| Hanner polytopes | Defined recursively: a segment, or the ℓ1 or ℓ∞ sum of two lower-dimensional Hanner polytopes[4](https://ar5iv.labs.arxiv.org/html/1212.2544) |
| Volume product | Every Hanner polytope has volume product 4^n/n!, the conjectured minimum among symmetric convex bodies[5](https://ar5iv.labs.arxiv.org/html/2203.13990) |
| Mahler conjecture | Confirmed for n = 2 and n = 3; open for n ≥ 4[5](https://ar5iv.labs.arxiv.org/html/2203.13990) |
| Publication profile | MathSciNet lists work in convex and discrete geometry (4 papers), topology (3 papers, 102 citations), and operations research (2 papers)[6](https://mathscinet.ams.org/mathscinet/MRAuthorID/191274) |
## Life and career
Hanner took his doctorate at [Stockholm University](https://www.edgepedia.org/stockholm-university) in 1952 with the dissertation *Retraction and extension of mappings*, published by Almqvist & Wiksell in Uppsala, under the advisor [Fritz Carlson](https://www.edgepedia.org/fritz-carlson).[3](https://mathgenealogy.org/id.php?id=20636) • [7](https://www.avhandlingar.se/avhandling/2bf24115da/) The French national library authority record likewise credits him as docteur en mathématiques at the faculty of science of Stockholm in 1952.[8](https://www.idref.fr/156022974) He was affiliated with the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study), and appears in the Swedish biographical directories *Vem är Vem?* (1962) and *Vem är det* (2001).
From 1963 to 1989 he held the professorship at the University of Gothenburg.[1](https://www.ne.se/uppslagsverk/encyklopedi/l%C3%A5ng/olof-hanner) The Mathematics Genealogy Project records two doctoral students: Thomas Gunnarsson (Chalmers Tekniska Högskola, 1978) and Carl Fant (Göteborgs universitet, 1980).[3](https://mathgenealogy.org/id.php?id=20636) His most frequent coauthors were Hans Rådström (3 joint papers), [Paul T. Bateman](https://www.edgepedia.org/paul-t-bateman) (2), and Victor L. Klee, Jr. (2).[6](https://mathscinet.ams.org/mathscinet/MRAuthorID/191274) The Swedish encyclopedia summarizes his published work as lying mainly in topology, though MathSciNet's classification of his output spans topology, convex and discrete geometry, and operations research and mathematical programming.[1](https://www.ne.se/uppslagsverk/encyklopedi/l%C3%A5ng/olof-hanner) • [6](https://mathscinet.ams.org/mathscinet/MRAuthorID/191274)
## The Hanner inequality
In 1936 Clarkson proved that the spaces Lp and lp are uniformly convex for p > 1. Hanner's note "On the uniform convexity of Lp and lp", published in *Arkiv för Matematik* volume 3, issue 3, pages 239–244, on 22 February 1956, set out to give the best possible modulus-of-convexity function for these spaces.[2](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full)
Hanner was candid about precursors. He recorded that the left-hand inequality of his Theorem 1 had been proved by Bærling at a seminar in Uppsala in 1945 but did not seem to be in print, and that the right-hand inequality had been proved by Clarkson and by Boas.[2](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full) The French authority record dates the equation known as Hanner's inequality to 1956.[8](https://www.idref.fr/156022974) For p = 2 the three terms of the inequality are equal for any x and y, recovering the parallelogram identity \( \|x+y\|^{2} + \|x-y\|^{2} = 2\|x\|^{2} + 2\|y\|^{2} \).[2](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full)
## Hanner polytopes
A symmetric convex body H is called a Hanner polytope if H is one-dimensional, or it is the ℓ1 or ℓ∞ sum of two lower-dimensional Hanner polytopes; equivalently, a Hanner polytope is the iterated ℓ1 or ℓ∞ sum of segments.[4](https://ar5iv.labs.arxiv.org/html/1212.2544) • [9](https://glivshyts6.math.gatech.edu/Fradelizi_slides.pdf)
The family has a combinatorial characterization: Hanner polytopes are in one-to-one correspondence with the perfect graphs that do not contain any induced path of edge length 3.[4](https://ar5iv.labs.arxiv.org/html/1212.2544) Their geometric significance comes from the Mahler volume product \( \mathcal{P}(K) = |K|\,|K^{\circ}| \), the product of the volumes of a convex body and its polar. Every Hanner polytope has the same volume product as the cube or the cross-polytope, namely \( 4^{n}/n! \), and each is a strict local minimizer of the volume product among symmetric convex bodies under the Banach–Mazur distance.[4](https://ar5iv.labs.arxiv.org/html/1212.2544) • [5](https://ar5iv.labs.arxiv.org/html/2203.13990) The cube's strict local minimality itself was first proved by Nazarov, Petrov, Ryabogin, and Zvavitch.[4](https://ar5iv.labs.arxiv.org/html/1212.2544)
## By the numbers
The two competing extremal values frame the subject. For origin-symmetric bodies the conjectured minimum of the volume product is
\[ \mathcal{P}(K) \geq \frac{4^{n}}{n!}, \]
attained by the cube and its polar the cross-polytope, and more generally by every Hanner polytope.[5](https://ar5iv.labs.arxiv.org/html/2203.13990) • [10](https://mathconjectures.com/conjectures/DG-007) For arbitrary convex bodies the conjectured bound is
\[ \mathcal{P}(K) \geq \frac{(n+1)^{n+1}}{(n!)^{2}}, \]
with equality if and only if K is a simplex.[5](https://ar5iv.labs.arxiv.org/html/2203.13990) Mahler proved the planar case of the symmetric problem and the non-symmetric case for n = 2; the symmetric case is confirmed for n = 2 and n = 3, and open for n ≥ 4.[5](https://ar5iv.labs.arxiv.org/html/2203.13990) • [11](https://arxiv.org/html/2605.13795v1) In three dimensions, equality in the symmetric bound holds if and only if K is a three-dimensional Hanner polytope, that is, the image of the regular octahedron or the cube by an invertible linear transformation.[12](https://arxiv.org/abs/2007.08736)
## How it compares with other extremal polytopes
The simplex, the cube, and the cross-polytope play parallel roles in the two halves of the Mahler problem: the simplex is the conjectured and proven-in-the-plane minimizer in the non-symmetric class, while the cube–cross-polytope pair anchors the symmetric class, with Hanner polytopes as the full conjectured equality family.[5](https://ar5iv.labs.arxiv.org/html/2203.13990) • [10](https://mathconjectures.com/conjectures/DG-007) The same family appears in a second extremal problem: Kalai's 3^d conjecture, likened to the Mahler conjecture, is also conjectured to be attained on the Hanner polytopes, and every one-dimensional centrally symmetric polytope is a Hanner polytope.[13](http://arxiv.org/pdf/2308.02909v2)
In dimension 3 there are precisely two GL(n,ℝ)-inequivalent Hanner polytopes: the cube [−1,1]^3 and its polar, the ℓ1 ball.[14](https://doi.org/10.2140/apde.2024.17.2179) This inequivalence matters for the conjecture's difficulty: the conjectured minimizers are nonunique in the strong sense that they lie in different GL(n,ℝ) orbits, and the same source proposes Lp-Mahler conjectures (for 0 < p < 1) under which the cube would be the unique minimizer among symmetric convex bodies.[14](https://doi.org/10.2140/apde.2024.17.2179)
## Recent developments
Work on the problems Hanner's polytopes anchor has continued. A 2026 arXiv paper proves the symmetric Mahler inequality in dimension three via admissible shadow systems, with equality attained precisely by affine images of the cube and of the cross-polytope.[11](https://arxiv.org/html/2605.13795v1) Kalai's 3^d conjecture is known to hold in dimensions d ≤ 4.[13](http://arxiv.org/pdf/2308.02909v2) A 2025 paper on Kalai's flag conjecture for locally anti-blocking polytopes uses the recursive definition of Hanner polytopes, noting they are the conjectured Mahler minimizers,[15](https://arxiv.org/pdf/2507.22284v2) and a 2026 preprint provides asymptotics for the face numbers of a certain family of Hanner polytopes, coming close to saturating the FLM inequality for certain parameters.[16](https://arxiv.symmetricfunctions.com/paper/2603.03861v1)
## References
1. [Olof Hanner, Uppslagsverk NE.se](https://www.ne.se/uppslagsverk/encyklopedi/l%C3%A5ng/olof-hanner)
2. [Olof Hanner, On the uniform convexity of Lp and lp, Arkiv för Matematik 3(3):239–244 (1956)](https://projecteuclid.org/journals/arkiv-for-matematik/volume-3/issue-3/On-the-uniform-convexity-of-Lp-and-lp/10.1007/BF02589410.full)
3. [Olof Hanner, The Mathematics Genealogy Project](https://mathgenealogy.org/id.php?id=20636)
4. [Minimal volume product near Hanner polytopes (arXiv:1212.2544)](https://ar5iv.labs.arxiv.org/html/1212.2544)
5. [Minimal volume product of convex bodies with certain discrete symmetries and its applications (arXiv:2203.13990)](https://ar5iv.labs.arxiv.org/html/2203.13990)
6. [Hanner, Olof, MathSciNet author profile](https://mathscinet.ams.org/mathscinet/MRAuthorID/191274)
7. [Retraction and extension of mappings, AVHANDLINGAR.SE](https://www.avhandlingar.se/avhandling/2bf24115da/)
8. [Hanner, Olof (1922–....), IdRef authority record, BnF](https://www.idref.fr/156022974)
9. [Volume product, polytopes and finite dimensional Lipschitz-free spaces (Fradelizi, lecture slides)](https://glivshyts6.math.gatech.edu/Fradelizi_slides.pdf)
10. [Mahler Volume-Product Conjecture, Math Conjectures (DG-007)](https://mathconjectures.com/conjectures/DG-007)
11. [The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems (arXiv:2605.13795)](https://arxiv.org/html/2605.13795v1)
12. [Minimal volume product of three dimensional convex bodies with various discrete symmetries (arXiv:2007.08736)](https://arxiv.org/abs/2007.08736)
13. [On the 3^d conjecture and Mahler volume (arXiv:2308.02909)](http://arxiv.org/pdf/2308.02909v2)
14. [Lp-polarity, Mahler volumes, and the isotropic constant (exa.ai mirror)](https://doi.org/10.2140/apde.2024.17.2179)
15. [Kalai's flag conjecture for locally anti-blocking polytopes (arXiv:2507.22284)](https://arxiv.org/pdf/2507.22284v2)
16. [Asymptotics for face numbers of certain Hanner polytopes, with applications (arXiv:2603.03861)](https://arxiv.symmetricfunctions.com/paper/2603.03861v1)
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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Analysts and PDE researchers › Banach space geometry specialists*
*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*
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