Local logarithmic Brunn-Minkowski holds for zonoids.
problem Logarithmic Brunn-Minkowski conjecture for zonoids
method Bochner method variant
result Local form of conjecture proven for zonoids
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11 results for “zonoids”
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.
problem Proving the log-Brunn-Minkowski inequality for convex bodies and zonoids.
method Establishing monotonicity of the deficit in the LLBM under line segment addition.
result Equality in LLBM for smooth convex bodies occurs only for homothetic bodies.
The main results are two characterisations of log-concave densities in terms of the collection of lift zonoids corresponding to a peacock. These notions are recalled and connected to arbitrage-free asset pricing in financial mathematics.
Solves approximation problems for zonoids and neural networks, closing gaps in dimensions 2 and 3.
problem Approximating zonoids and shallow neural networks in uniform norm.
method Combines techniques to solve both problems, closing gaps in dimensions 2 and 3.
result Completes the solution for zonoid approximation in all dimensions and improves neural network approximation rates.
Probabilistic theory counts intersections in Riemannian spaces.
problem Counting intersections in Riemannian homogeneous spaces.
method Introduces probabilistic intersection ring , a graded commutative and associative real Banach algebra.
result Probabilistic intersection ring structure defined for spheres, real projective space, and complex projective space.
The space of call price functions has a natural noncommutative semigroup structure with an involution. A basic example is the Black--Scholes call price surface, from which an interesting inequality for Black--Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore …
We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk meas…
A new procedure, called DDa-procedure, is developed to solve the problem of classifying d-dimensional objects into q >= 2 classes. The procedure is completely nonparametric; it uses q-dimensional depth plots and a very efficient algorithm for discrimination analysis in the depth space [0,1]^q. Specifically, the depth i…
In this paper we show how to relate European call and put options on multiple assets to certain convex bodies called lift zonoids. Based on this, geometric properties can be translated into economic statements and vice versa. For instance, the European call-put parity corresponds to the central symmetry property, while…
Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transportstat.ML
Improved persistence spheres map measures to functions, stable under partial transport.
problem Representing and comparing measures in topological machine learning.
method Persistence spheres map measures to continuous functions on the sphere, stable under 1-Wasserstein partial transport.
result Persistence spheres provide a stable, parameter-free representation of measures, improving upon existing methods.