## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the *Duke Mathematical Journal* is one of the world's leading mathematical journals. *DMJ* emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

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**ISSN:**0012-7094 (print), 1547-7398 (electronic)**Publisher:**Duke University Press**Discipline(s):**Mathematics**Full text available in Euclid:**1935--**Access:**By subscription only**Euclid URL:**http://projecteuclid.org/dmj

### In memoriam

The *Duke Mathematical Journal* announces with sadness the passing of
Duke Emeritus Professor Morris Weisfeld on 8 April 2017. Morris was Managing Editor of *DMJ* for all but 18 months of the period 1973–1997, during which time he increased its annual size from 1,000 to 3,000 pages. Under his leadership, *DMJ* became a much more interesting journal, with an eclectic mix of papers that contrasted (at the time) with *Annals* and *Inventiones*. More significantly, it was
through his commitment to excellence that *DMJ* became one of the world’s
most highly regarded mathematical journals.

Jonathan Wahl

Managing Editor, *DMJ*

### Featured bibliometrics

*MR Citation Database* MCQ (2015): 2.29

*JCR* (2015) Impact Factor: 2.350

*JCR* (2015) Five-year Impact Factor: 2.337

*JCR* (2015) Ranking: 9/312 (Mathematics)

Article Influence: 3.899

Eigenfactor: Duke Mathematical Journal

SJR/SCImago Journal Rank (2015): 5.675

Indexed/Abstracted in: *Current Contents: Physical, Chemical and Earth Sciences*, *IBZ Online*, *Magazines for Libraries*, *MathSciNet*, *Science Citation Index*, *Science Citation Index Expanded*, *Scopus*, and *zbMATH*

### Featured article

* *Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $PSL(3,\mathbb{R})$

##### Abstract

This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}{\mathbf{P}}^{n}$. These domains carry several projective invariant distances, among which are the Hilbert distance ${d}^{H}$ and the Blaschke distance ${d}^{B}$. We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain,

$${d}^{B}(x,y)<{d}^{H}(x,y)+1.$$

We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}{\mathbf{P}}^{n}$, the volume of a ball of radius $R$ grows at most like ${e}^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation $\rho $ of a surface group $\Gamma $ into $PSL(3,\mathbb{R})$, there exists a Fuchsian representation $j:\Gamma \to PSL(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than that of $\rho $. This answers positively a conjecture of Lee and Zhang in the $3$-dimensional case.