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Compact convex sets of the plane and probability theory

Jean-François Marckert, David Renault (2014)

ESAIM: Probability and Statistics

The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ∫ 0 2 π e i x d μ ( x ) = 0 ∫ 0 2 π e ix d μ ( x ) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS – for example, the Minkowski sum – have natural translations in terms of probability measure operations,...

Convex hulls, Sticky particle dynamics and Pressure-less gas system

Octave Moutsinga (2008)

Annales mathématiques Blaise Pascal

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities u 0 with negative jumps. We show the existence of a stochastic process and a forward flow φ satisfying X s + t = φ ( X s , t , P s , u s ) and d X t = E [ u 0 ( X 0 ) / X t ] d t , where P s = P X s - 1 is the law of X s and u s ( x ) = E [ u 0 ( X 0 ) / X s = x ] is the velocity of particle x at time s ≥ 0 . Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map ( x , t ) ↦ M ( x , t ) : = P ( X t ≤ x ) is the entropy solution of a scalar conservation law ∂ t M + ∂ x ( A ( M ) ) = 0 where the flux A represents the particles...

Convex sets and inequalities.

Takahasi, Sin-Ei, Takahashi, Yasuji, Miyajima, Shizuo, Takagi, Hiroyuki (2005)

Journal of Inequalities and Applications [electronic only]

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