In this paper is discussed an application of signed measures (charges) to description of segment and chord length distributions in nonconvex bodies. The signed distribution may naturally appears due to definition via derivatives of nonnegative autocorrelation function simply related with distances distribution between pairs of points in the body. In the work is suggested constructive geometrical interpretation of such derivatives and illustrated appearance of "positive" and "negative" elements similar with usual Hanh-Jordan decomposition in measure theory. The construction is also close related with applications of Dirac method of chords.
Different properties of chord length distribution (CLD) for nonconvex bodies were discussed recently in few publications [1,2,3,4,5,6,7]. Let us recall three different ways to introduce CLD for nonconvex body. Straight line may intersect nonconvex body more than one time (see Fig. 1) and we can either consider each segment of such line as separate chord or calculate sum of all such segments. These two methods are known as multi-chord and one-chord distribution (MCD and OCD) respectively [1,2,3]. For the convex case probability density function for CLD is proportional to second derivative of autocorrelation function [1,4,5,6]. Such a property may be used for a third definition of "the generalized chord distribution" [4,5,6], but straightforward calculations demonstrates possibility of negativity of such function for some nonconvex bodies [6]. Let us call this function here signed chord (length) distribution to avoid some ambiguity of term "generalized" and to emphasize the basic distinguishing property of this function.
Definition of CLD for convex body has standard probabilistic interpretation in theory of geometric probability and random sets [8,9,10]. The MCD and OCD cases for nonconvex body may be described as well [1,2,3]. Is it possible to consider similar possibility for the signed chord distribution?
In wonderful essay “Negative probability” [11] Feynman wrote that, unlike “final probability of a verifiable physical event”, “conditional probabilities and probabilities of imagined intermediate states may be negative” and so: “If a physical theory for calculating probabilities yields a negative probability for a given situation under certain assumed conditions, we need not conclude the theory is incorrect.” In this review Feynman provided a few examples with appearance and interpretation of negative probabilities both for quantum and classical physical models.
Mathematical extension of the measure theory for such a purposes may use so-called signed measures (charges) [12]. Usually such extension is reduced to standard positive measures due to Hahn and Jordan decompositions, corresponding to expression of charge as difference of two positive measures [12].
In many processes with signed distributions the Hahn decomposition, i.e., splitting of space of events on positive and negative parts is quite obvious, e.g., in simplest examples we have two kinds of events: putting and removing objects [11]. A distinction of the signed chord distribution is appearance of negativity due to differentiations of positive function without such a natural decomposition on positive and negative elements.
Nonconvex body Fig. 1b may be represented as a convex body and a convex hole and it provides some intuitive justification of possibility to express some distributions using formal difference of convex hull and the hole. Rigor consideration is more difficult, especially for chord distribution expressed via second derivative, e.g., method derived below in Sec. The convex case is revisited in Sec. 2 and Appendix A. The construction of signed chord length distribution for nonconvex body is described in Sec. 3. Some implications to description of arbitrary bodies with nonuniform density are briefly mentioned for completeness in Sec. 4 and Appendix B. Other extensions, like polygonal trajectories are affected very shortly in Sec. 5 and Appendix C.
There are many different functions and relations between them used for description properties of convex bodies [1,2,3,4,5,6,7,8,9,10,14,15,16]. In present paper are considered three different kinds of distributions: distances between points Fig. 2a, lengths of radii (segments) Fig. 2b, and lengths of chords Fig. 2c. It may be useful to describe precisely models of generation of each distribution, to avoid some problems with ambiguity, similar with widely known Bertrand paradox [8,13]. It is also convenient to use autocorrelation function γ(l) related with distances distribution for three-dimenional body as
where V is volume of V. The expression for autocorrelation function for body with arbitrary density Eq. (A5) together with derivation of Eq. (A8) for constant density is recollected for completeness below in Appendix A-2.
There is remarkable correspondence between these densities [1,2,3,4,5,6,15,16]:
where average chord length l = ∞ 0 lµ(l)dl may be expressed via volume V and surface area S using a relation for three-dimensional convex body derived in XIX century by Cauchy, Czuber and rediscovered later by Dirac et al [8,10,14,15]
Distribution of lengths of radii [16] is also known as interior source randomness [15]. Relation between µ(l) and ι(l) in Eq. (2.1) is often represented in integral form [15,16]
Proportionality between µ(l) and second derivative of autocorrelation function γ(l) in Eq. (2.1) is also well known and widely used [1,4,5,6].
Here is convenient for completeness of presentation and further explanation of nonconvex case to derive all equalities in Eq. (2.1
This content is AI-processed based on open access ArXiv data.