Experimental investigation on the uniqueness of a center of a body

Abstract

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in R m . When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called an r α−m -center or an illuminating center, respectively.The existence of a center is easily shown but the uniqueness does not always hold. Sufficient conditions of the uniqueness of a center have been studied by some researchers. The main results in this paper are some new sufficient conditions for the uniqueness of a center of a body. , 26B25. points of the line segments OP , P Q and QO, respectively. We remark that the minimal unfolded region of Ω is contained in △ABC (see Example 2.5).We identify the notation z j for the j-th coordinate with the function z j : R 2 ∋ (z 1 , z 2 ) → z j ∈ R. We denote the point R θ P by P θ , for short, and so on.We have to consider the following eleven cases about the position of R θ Ω (see Figure 9 to 19):Case I The rotation angle θ is non-negative.and the slope of the line P θ Q θ is non-positive. Case I.3.2 0 ≤ z 1 (B θ ) ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-negative. Case I.4.1 z 1 (B θ ) ≤ 0 ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-positive. Case I.4.2 z 1 (B θ ) ≤ 0 ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-negative.Case II The rotation angle θ is non-positive.Case II.2.1 z 1 (A θ ) ≤ z 1 (C θ ) ≤ z 1 (P θ ) and the slope of the line OQ θ is non-negative.Case II.2.2 z 1 (A θ ) ≤ z 1 (C θ ) ≤ z 1 (P θ ) and the slope of the line OQ θ is non-positive.Case II.3.1 z 1 (P θ ) ≤ z 1 (C θ ) and the slope of the line OQ θ is non-negative.Case II.3.2 z 1 (P θ ) ≤ z 1 (C θ ) and the slope of the line OQ θ is non-positive.

Stationary radial centers and symmetry of convex polytopes

“…Thanks to Lemma 2.4, ∇V Ω . Some sufficient conditions for the uniqueness were studied in, for example, [8,19,22].…”

Geometric Estimation of a Potential and Cone Conditions of a Body

“…We refer to [6] for the physical meaning of the study on centers of a body. The uniqueness of a radial center was discussed in [7,13] but the investigation in [7] has an error, and it was pointed out in [17,20]. is strictly concave on Ω.…”

“…Using the radial symmetry of the kernels of the potentials mentioned in the previous subsections, we can restrict the region containing those centers. We introduce the restricted region and its properties from [2,17,19] (see also [1,18,20]). Put Figure 2).…”

“…We refer to [1,2,17,18,19,20] for the study on the location of maximizers of a potential. Some authors tried to restrict the location of maximizers of a potential into a smaller region.…”

“…This is because we can use the expression (1.10) of the potential V (−1) Ω only in the interior of Ω. Namely, in (1.10), we want to take a uniform ε for illuminating centers of any small enough height and maximizers of V (−1) Ω . We refer to [1,2,17,18,19,20] for the study on the location of maximizers of a potential. Some authors tried to restrict the location of maximizers of a potential into a smaller region.…”