scholarly journals Symmetric convex sets with minimal Gaussian surface area

2021 ◽  
Vol 143 (1) ◽  
pp. 53-94
Author(s):  
Steven Heilman
Keyword(s):  
Surface Area ◽  
Convex Sets ◽  
Symmetric Convex ◽  
Gaussian Surface

On the Origin of Four Pi in Physics

2016 ◽  
pp. 3994-4013
Author(s):  
Aaron Hanken
Keyword(s):  
Black Hole ◽  
Surface Area ◽  
Single Particles ◽  
Pythagorean Triples ◽  
Free Particles ◽  
Close Relationship ◽  
Dimensional Parameters ◽  
Gaussian Surface ◽  
Planck Energy ◽  
The Universe

We find the highest symmetry between the fields intrinsic to free particles (free particles having only mass, charge and spin), and show these fields symmetries and their close relationship to force and entropy. The Boltzmann Constant is equal to the natural entropy, in that it is The Planck Energy over The Planck Temperature. This completes a needed symmetry in The Bekenstein-Hawking Entropy. Upon substitution of Planck Units into The Schwarzschild Radius, we find that the mass and radius of any black hole define both the gravitational constant and the natural force. We find that the Gaussian Surface area about a particle is equal to the surface area of an equally massed black hole if we define the gravitational field of that particle to be the quotient of The Planck Force and the particles mass. By these simple substitutions we find that gravity is quantized in units of surface entropy. We also find Pythagorean Triples are resting within the dimensional parameters of Special Relativity, and show this to be the dimensional aspects of single particles observing one another, coupled with the intrinsic Hubble nature of the universe.

scholarly journals Helly-Type Theorems in Property Testing

2018 ◽  
Vol 28 (04) ◽  
pp. 365-379
Author(s):  
Sourav Chakraborty ◽  
Rameshwar Pratap ◽  
Sasanka Roy ◽  
Shubhangi Saraf
Keyword(s):  
High Probability ◽  
Convex Sets ◽  
Property Testing ◽  
Geometric Object ◽  
Fundamental Result ◽  
Symmetric Convex ◽  
Helly’S Theorem ◽  
Convex Object ◽  
Constant Size ◽  
Testing Algorithm

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If [Formula: see text] is a set of [Formula: see text] points in [Formula: see text], we say that [Formula: see text] is [Formula: see text]-clusterable if it can be partitioned into [Formula: see text] clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object [Formula: see text]. In this paper, as an application of Helly’s theorem, by taking a constant size sample from [Formula: see text], we present a testing algorithm for [Formula: see text]-clustering, i.e., to distinguish between the following two cases: when [Formula: see text] is [Formula: see text]-clusterable, and when it is [Formula: see text]-far from being [Formula: see text]-clusterable. A set [Formula: see text] is [Formula: see text]-far [Formula: see text] from being [Formula: see text]-clusterable if at least [Formula: see text] points need to be removed from [Formula: see text] in order to make it [Formula: see text]-clusterable. We solve this problem when [Formula: see text], and [Formula: see text] is a symmetric convex object. For [Formula: see text], we solve a weaker version of this problem. Finally, as an application of our testing result, in the case of clustering with outliers, we show that with high probability one can find the approximate clusters by querying only a constant size sample.

On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

2002 ◽  
Vol 53 (3-4) ◽  
pp. 245-248
Author(s):  
Subir K. Bhandari ◽  
Ayanendranath Basu
Keyword(s):  
Recent Result ◽  
Convex Sets ◽  
Convex Set ◽  
Correlation Inequalities ◽  
The Other ◽  
Symmetric Convex ◽  
Complete Proof ◽  
Other Hand

Pitt's conjecture (1977) that P( A ∩ B) ≥ P( A) P( B) under the Nn (0, In) distribution of X, where A, B are symmetric convex sets in IRn still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRn. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitt's conjecture.

scholarly journals Some inequalities for symmetric convex sets with applications

1996 ◽  
Vol 24 (2) ◽  
pp. 753-762 ◽  
Author(s):  
T. W. Anderson
Keyword(s):  
Convex Sets ◽  
Symmetric Convex

scholarly journals Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

2018 ◽  
Vol 24 (2) ◽  
pp. 479-494
Author(s):  
Davit Harutyunyan
Keyword(s):  
Convex Sets ◽  
Space Dimension ◽  
Arithmetic Mean ◽  
Symmetric Convex ◽  
Mass Transportation ◽  
Best Constant ◽  
Centrally Symmetric

In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constant C(n) = Cn7 depending on the space dimension n in both inequalities is due to Segal [A. Segal, Lect. Notes Math., Springer, Heidelberg 2050 (2012) 381–391]. We improve that constant to Cn6 for convex sets and to Cn5 for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form Cn2, i.e., quadratic in n. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

scholarly journals On the area of planar convex sets containing many lattice points

1987 ◽  
Vol 35 (3) ◽  
pp. 441-454
Author(s):  
P. R. Scott
Keyword(s):  
Large Class ◽  
Convex Sets ◽  
Convex Set ◽  
Lattice Points ◽  
Symmetric Convex ◽  
Classical Theorem ◽  
Planar Convex

A classical theorem of van der Corput gives a bound for the volume of a symmetric convex set in terms of the number of lattice points it contains. This theorem is here generalized and extended for a large class of non-symmetric sets in the plane.

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