On the Uniqueness of Gravitational Centre

2020

Colloq. Math.

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

Section: Balance Law For Riesz Potentialsmentioning

confidence: 99%

“…Thus every r α−m -center coincides with the critical point of V Ω . Some sufficient conditions for the uniqueness were studied in, for example, [8,19,22].…”

Section: Balance Law For Riesz Potentialsmentioning

confidence: 99%

Stationary radial centers and symmetry of convex polytopes

2020

Colloq. Math.

“…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.…”

Section: Renormalization Of the Riesz Potentialmentioning

confidence: 99%

Geometric Estimation of a Potential and Cone Conditions of a Body

2017

J Geom Anal

We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, we give an estimate of the potential using the exterior cones of the body. Moreover, we apply the result to the Poisson integral for the upper half space.

“…In [13], when φ(ρ) = ρ α in (1.2), Moszyńska called a maximum point of Φ A a radial center of order α and showed that if 0 < α ≤ 1, then every convex body has a unique radial center of order α. In [8], for α > 1, the uniqueness of a radial center of a convex body was studied but the argument included an error. In [9], Herburt studied the location of a radial center of order 1.…”

Section: Introductionmentioning

confidence: 99%

Experimental investigation on the uniqueness of a center of a body

2016

Ann. Polon. Math.

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.