Geodesic
Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with $SO(k)\times SO(n-k)$-symmetry
Canadian journal of mathematics, Tome 74 (2022) no. 5, pp. 1486-1516

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For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$, we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$. Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$-symmetry.
DOI : 10.4153/S0008414X21000304
Mots-clés : Minkowski norm, Hessian isometry, Laugwitz Conjecture, Landsberg Unicorn Conjecture, Legendre transformation 
Xu, Ming; Matveev, Vladimir S. Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with $SO(k)\times SO(n-k)$-symmetry. Canadian journal of mathematics, Tome 74 (2022) no. 5, pp. 1486-1516. doi: 10.4153/S0008414X21000304
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     author = {Xu, Ming and Matveev, Vladimir S.},
     title = {Proof of {Laugwitz} {Conjecture} and {Landsberg} {Unicorn} {Conjecture} for {Minkowski} norms with $SO(k)\times SO(n-k)$-symmetry},
     journal = {Canadian journal of mathematics},
     pages = {1486--1516},
     year = {2022},
     volume = {74},
     number = {5},
     doi = {10.4153/S0008414X21000304},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X21000304/}
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