In this paper, we study the Dirichlet problem associated with infinity-Laplacian type equations that may exhibit anisotropic character. We identify a broad class of nonlinearities for which the ...problem may or may not admit viscosity solutions for any continuous boundary data. We also discuss comparison with Finsler cones, which may be of independent interest.
We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for ...the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.
We describe the basic properties of a norm and introduce the Minkowski norm. We then show that the OWA aggregation operator can be used to provide norms. To enable this we require that the OWA ...weights satisfy the buoyancy property, w j ¿ w k for j < k . We consider a number of different classes of OWA norms. It is shown that the functional generation of the weights of an OWA norm requires the weight generating function have a non-positive second derivative. We discuss the use of the generalized OWA operator to provide norms. Finally we describe the use of OWA operators to induce similarity measures.
Let
M
t
be an isoparametric foliation on the unit sphere (
S
n
−1
(1),
g
st
) with
d
principal curvatures. Using the spherical coordinates induced by
M
t
, we construct a Minkowski norm with the ...representation
F
=
r
2
f
(
t
)
, which generalizes the notions of (
α, β
)-norm and (
α
1
,
α
2
)-norm. Using the technique of the spherical local frame, we give an exact and explicit answer to the question when
F
=
r
2
f
(
t
)
really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry Φ between two Minkowski norms induced by
M
t
, which preserves the orientation and fixes the spherical
ξ
-coordinates. There are two ways to describe this Φ, either by a system of ODEs, or by its restriction to any normal plane for
M
t
, which is then reduced to a Hessian isometry between Minkowski norms on ℝ
2
satisfying certain symmetry and (d)-properties. When
d
> 2, we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications, so it must satisfy the (d)-property for any orthogonal decomposition ℝ
n
=
V
′ +
V
″, i.e., for any nonzero
x
=
x
′ +
x
″ and
Φ
(
x
)
=
x
¯
=
x
¯
′
+
x
¯
′
′
with
x
′
,
x
¯
′
∈
V
′
and
x
″,
x
′
′
,
x
¯
′
′
∈
V
′
′
, we have
g
x
F
1
(
x
′
′
,
x
)
=
g
x
¯
F
2
(
x
¯
′
′
,
x
¯
)
. As byproducts, we prove the following results. On the indicatrix (
S
F
,
g
), where
F
is a Minkowski norm induced by
M
t
and
g
is the Hessian metric, the foliation
N
t
=
S
F
∩ ℝ
>0
M
0
is isoparametric. Laugwitz Conjecture is valid for a Minkowski norm
F
induced by
M
t
, i.e., if its Hessian metric
g
is flat on ℝ
n
{0} with
n
> 2, then
F
is Euclidean.
Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit ...spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds Wagner (Dokl Acad Sci USSR (N.S.) 39:3–5, 1943). Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics Bartelmeß and Matveev (J Diff Geom Appl 58:264–271, 2018), extremal compatible linear connections and algorithmic solutions Vincze (Aequat Math 96:53–70, 2022)), it is very hard to solve them in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved Vincze (Publ Math Debrecen 83(4):741–755, 2013 (see also Vincze (Euro J Math 3:1098–1171, 2017))) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the uniqueness by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities.
In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, ...we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.
In this note we prove some Rogers–Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard ...trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.
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