Publication

On extremal problems associated with random chords on a circle

2025

Abstract

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius , where the endpoints of the chords are drawn according to a given probability distribution on . We show that, for , the problem is degenerated in the sense that any continuous measure is an extremizer, and that, for sufficiently close to 1, the desired maximal value is strictly below the one for by a polynomial factor in . Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is for . Connections with other variational problems and energy minimization problems are also presented.

Official source

https://infoscience.epfl.ch/entities/publication/fdaca089-574c-4e0a-916a-8c59e62d2aa4

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2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/fdaca089-574c-4e0a-916a-8c59e62d2aa4

About this result

Related concepts (32)

Probability distribution

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

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On extremal problems associated with random chords on a circle

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