For given T > 0 and n > 0, are there any results that characterize curves x : [0,T] -> Rn \ {0} satisfying \int_{0}^{T} \frac{ xi(s)}{ || x(s) ||} \frac{ xj(s)}{|| x(s) || } ds = (T/2) dij, where x = (x1, ....,xn), "d" is the Kronecker delta function and || * || is the Euclidean norm.
This is a good question with lots of paths to follow in looking for answers. A promising place to start with is the characterisation of curves of constant geodesic curvature. See, for example, Section 3.5, starting on page 38, in
Z. Langi, Convexity Problems in Spaces of Constant Curvature and in Normed Spaces, Ph.D. thesis, The University of Calgary, 2008:
http://www.math.bme.hu/~zlangi/publications/formalthesis.pdf
I like this approach because it takes into account convexity in the form of convex sets, convex bodies, and convex polytopes, starting on page 8. Later, spindle convex sets are considered, starting on page 68.
Another excellent place to look is in the survey article by H. Martini:
H. Martini, K.J. Swanepoel, The Geometry of Minkowski Spaces--A Survey. Part II, Expostiones Mathematicae 22(2), 2004, 93-144.
http://www.sciencedirect.com/science/article/pii/S0723086904800094
This is a 50 page article with lots of interesting results. Again, generalised convexity notions are considered in this article, starting on page 94.
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