The distance between the second and first eigenvalues for the Dirichlet Laplacian of a domain is called its (spectral) gap. We show that the gap of a convex planar domain$D$symmetric about both the$x$and$y$axes is no smaller than the gap of an oriented rectangle which contains$D$.

Let$f=g$_{t}+$h$_{t}be the optimal decomposition for calculating the exact value of the$K$-functional$K(t, f$; $$\bar X$$ ) of an element$f$with respect to a couple $$\bar X$$ =($X$_{0},$X$_{1}) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spaces$X$_{0}and$X$_{1}is either$L$^{∞}or$L$^{1}. Many examples are given of couples of lattices $$\bar X$$ for which |$g$_{t}| increases monotonically a.e. with respect to$t$. It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak$K$-divisibility constant γ( $$\bar X$$ ) for the couple. But it is also shown that certain couples $$\bar X$$ do not have this property. These also provide examples of couples of lattices for which γ( $$\bar X$$ ).

Let Ω be a bounded convex domain in C_{$n$}, with smooth boundary of finite type$m$.

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The subject of this paper is a Jacobian, introduced by F. Lazzeri (unpublished), associated with every compact oriented Riemannian manifold whose dimension is twice an odd number. We study the Torelli and Schottky problem for Lazzeri's Jacobian of flat tori and we compare Lazzeri's Jacobian of Kähler manifolds with other Jacobians.