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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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