Analogues of the Markov and Bernstein inequalities on convex bodies in Banach spaces

Abstract

A 6-dimensional grand unified theory with the compact space having the topology of a real projective plane, i.e., a 2-sphere with opposite points identified, is considered. The space is locally flat except for two conical singularities where the curvature is concentrated. One supersymmetry is preserved in the effective 4d theory. The unified gauge symmetry, for example SU(5) , is broken only by the non-trivial global topology. In contrast to the Hosotani mechanism, no adjoint Wilson-line modulus associated with this breaking appears. Since, locally, SU(5) remains a good symmetry everywhere, no UV-sensitive threshold corrections arise and SU(5)-violating local operators are forbidden. Doublettriplet splitting can be addressed in the context of a 6d N = 2 super Yang-Mills theory with gauge group SU(6). If this symmetry is first broken to SU(5) at a fixed point and then further reduced to the standard model group in the above non-local way, the two light Higgs doublets of the MSSM are predicted by the group-theoretical and geometrical structure of the model.

A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

“…Also, another real, geometric method, of obtaining Bernstein type inequalities, due to Skalyga [39,40], is to be mentioned here: the difficulty with that is that to the best of our knowledge, no one has ever been able to compute, neither for the seemingly least complicated case of the standard triangle of R 2 , nor in any other particular non-symmetric case the yield of that abstract method. Hence in spite of some remarks that the method is sharp in some sense, it is unclear how close these estimates are to the right answer and what use of them we can obtain in any concrete cases.…”

Bernstein and Markov-type inequalities for polynomials on real Banach spaces

“…(i) Skalyga [28,29] has obtained estimates on the derivatives of polynomials defined on real Banach spaces. For higher derivatives Skalyga's estimates are not optimal.…”