On the Asymmetry for Convex Domains of Constant Width

The extremal convex bodies of constant width for the Minkowski measure of asymmetry are discussed. A result, similar to that of H. Groemer＇s and of H. Lu＇s, is obtained, which states that, for the Minkowski measure of asymmetry, the most asymmetric convex domains of constant width in R2 are Reuleaux triangles.

Portfolio diversification benefits of alternative currency investment in Bitcoin and foreign exchange markets

This study examines the portfolio diversification benefits of alternative currency trading in Bitcoin and foreign exchange markets.The following methods are applied for the analysis:the spillover index method of Diebold and Yilmaz(Int J Forecast 28(1):57–66,2012.https://doi.org/10.1016/j.ijfor ecast.2011.02.006),the spillover asymmetry measures of Barunik et al.(J Int Money Finance 77:39–56,2017.https://doi.org/10.1016/j.jimon fin.2017.06.003),and the frequency connectedness method of Barunik and Křehlik(J Financ Econom 16(2):271–296,2018.https://doi.org/10.1093/jjfin ec/nby001).The findings identify the presence of low-level integration and asymmetric volatility spillover as well as a dominant role of short horizon spillover among Bitcoin markets and foreign exchange pairs for six major trading currencies(US dollar,euro,Japanese yen,British pound sterling,Australian dollar,and Canadian dollar).Bitcoin is found to provide significant portfolio diversification benefits for alternative currency foreign exchange portfolios.Alternative currency Bitcoin trading in euro is found to provide the most significant portfolio diversification benefits for foreign exchange portfolios consisting of major trading currencies.The findings of the study regarding spillover dynamics and portfolio diversification capabilities of the Bitcoin market for foreign exchange markets of major trading currencies have significant implications for portfolio diversification and risk minimization.

Approximation of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.

On Properties of p-critical Points of Convex Bodies

Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on （1, ＋∞） is confirmed, and the connections between general p-critical points and the Minkowski-critical points （∞-critical points） are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.

Stochastic Maximum Principle for Optimal Control of Forward-backward Stochastic Pantograph Systems with Regime Switching

We introduce a family of measures（functions） of asymmetry for convex bodies and discuss their properties. It turns out that this family of measures shares many nice properties with the mean Minkowski measures. As the mean Minkowski measures describe the symmetry of lower dimensional sections of a convex body, these new measures describe the symmetry of lower dimensional orthogonal projections.

Mixed Volumes and Measures of Asymmetry

The mixed volume and the measure of asymmetry for convex bodies are two important topics in convex geometry.In this paper,we first reveal a close connection between the Lp-mixed volumes proposed by E.Lutwak and the p-measures of asymmetry,which have the Minkowski measure as a special case,introduced by Q.Guo.Then,a family of measures of asymmetry is defined in terms of the Orlicz mixed volumes introduced by R.J.Gardner,D.Hug and W.Weil recently,which is an extension of the p-measures.

The Minkowski Measure of Asymmetry for Spherical Bodies of Constant Width

In this paper,we introduce the Minkowski measure of asymmetry for the spherical bodies of constant width.Then we prove that the spherical balls are the most symmetric bodies among all spherical bodies of constant width,and the completion of the spherical regular simplexes are the most asymmetric bodies.

p-Minkowski Type Measures of Asymmetry for Convex Bodies

In this paper,the p-Minkowski type measures of asymmetry for convex bodies,which have the well-known Minkowski measure of asymmetry as the special case p=∞,are introduced,and some properties of the p-Minkowski type measures are investigated.

Asymmetry for Convex Body of Revolution

In this paper, we study the Minkowski measure of asymmetry for n-dimensional convex bodies of revolution （ n ≥ 3 ）. We show that among all n-dimensional convex bodies of revolution, the bodies which generated by isosceles triangles are the most asymmetric ones. Also, we study the asymmetry for n-dimensional constant width bodies of revolution.

Critical Chords of Convex Bodies of Constant Width

In this paper, we show that when Minkowski measure of asymmetry of convex body K of constant width is bigger than a（n-1）, K has at least n＋1 critical chords, where a（n）=n＋（2n（n＋1））√1/2/n＋2.