134,383 research outputs found
Convexity of Momentum Maps: A Topological Analysis
The Local-to-Global-Principle used in the proof of convexity theorems for
momentum maps has been extracted as a statement of pure topology enriched with
a structure of convexity. We extend this principle to not necessarily closed
maps f\colon X\ra Y where the convexity structure of the target space
need not be based on a metric. Using a new factorization of , convexity of
the image is proved without local fiber connectedness, and for arbitrary
connected spaces .Comment: 21 pages LaTeX2e; minor revisions, to appear in Topology and its
Applications; Dedicated to Alan D. Weinstein, Dennis P. Sullivan, and in
memory of Jerrold E. Marsden. arXiv admin note: substantial text overlap with
arXiv:1009.252
Generalized Convexity and Inequalities
Let R+ = (0,infinity) and let M be the family of all mean values of two
numbers in R+ (some examples are the arithmetic, geometric, and harmonic
means). Given m1, m2 in M, we say that a function f : R+ to R+ is
(m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in R+ . The
usual convexity is the special case when both mean values are arithmetic means.
We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient
conditions for (m1,m2)-convexity of functions defined by Maclaurin series. The
criteria involve the Maclaurin coefficients. Our results yield a class of new
inequalities for several special functions such as the Gaussian hypergeometric
function and a generalized Bessel function.Comment: 17 page
Convex Multivariable Trace Functions
For any densely defined, lower semi-continuous trace \tau on a C*-algebra A
with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function
f of n variables, we give a short proof of the fact that the function (x_1,
x_2, ..., x_n) --> \tau (f(x_1, x_2, ..., x_n)) is convex on the space
\bigoplus_{i=1}^n (A_i)_{self-adjoint}. If furthermore the function f is
log-convex or root-convex, so is the corresponding trace function. We also
introduce a generalization of log-convexity and root-convexity called
\ell-convexity, show how it applies to traces, and give a few examples. In
particular we show that the trace of an operator mean is always dominated by
the corresponding mean of the trace values.Comment: 13 pages, AMS TeX, Some remarks and results adde
Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
We prove convergence results for expanding curvature flows in the Euclidean
and hyperbolic space. The flow speeds have the form , where and
is a positive, strictly monotone and 1-homogeneous curvature function. In
particular this class includes the mean curvature . We prove that a
certain initial pinching condition is preserved and the properly rescaled
hypersurfaces converge smoothly to the unit sphere. We show that an example due
to Andrews-McCoy-Zheng can be used to construct strictly convex initial
hypersurfaces, for which the inverse mean curvature flow to the power
loses convexity, justifying the necessity to impose a certain pinching
condition on the initial hypersurface.Comment: 18 pages. We included an example for the loss of convexity and
pinching. In the third version we dropped the concavity assumption on F.
Comments are welcom
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