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Among the works on this topic, in [RT] the authors proved that for each n there is a constant K n such that c n X K n for every Banach space X. Enable Javascript to interact with content and submit forms on Wolfram Alpha websites. In order to do this we will work with measures satisfying a not too restrictive property.

The real case is similar and its proof can be found in Lemma. The author also presented a similar question to Tarski s Plank Problem. Insomecase, likeinthel p spacesandtheschattenclassess p, we obtain optimal lower bounds, while for other spaces we only give some estimates of desihualdades optimal lower bounds.

For multilinear operators between two spaces, we will use T and reserve the letters P and Q for polynomials. The relative width of a plank is the width of the plank divided by the width of the convex body in the direction that the plank attains its width. A prime example of this is the article [DM], where the authors give several relations between a variety of norms and the Mahler measure.

Another issue with this method is that finding a set of functions such that we desiyualdades equality in. These type of inequalities have been widely studied by several authors in a variety of contexts.

Now we are ready to state our method to obtain lower estimates of c x. The authors found its exact value and proved that, when the dimension d is large, the order of this constant is d.

In the proof of Proposition The later will be used mostly when we have both situations: The factor problem is the problem of finding lower bounds for the norm of the product of polynomials of some prescribed degrees. The authors found the exact value of c r d and proved that its order is d. We will treat separately the lower and the upper bound. In Chapter 3 we study the factor problem on several spaces. As pointed out before, a corollary from this result is that c n X n n for any complex Banach space.

When we consider a d-dimensional real or complex Hilbert desiguqldades H, taking in Theorem. Unfortunately, this stronger result is not necessarily valid if we choose other sets of unit functionals.

With the same argument we may assume that there is x 0 S X such that x n x 0. In the particular case when X is a Hilbert space, due desigusldades the symmetry of the sphere, we may even expect that they are uniformly distributed across the sphere.

Before proceeding with the proof, let us set some notation to lighten up the writing. All the Banach spaces considered will be either over the complex field C or the real field R, we write K when we mean either. Daniel Carando Lugar de trabajo: This, and the fact that the Mahler measure is multiplicative, makes it possible to deduce inequalities regarding the norm or the length of the product of polynomials using the Mahler measure as a tool.

More details on the Mahler measure can be found in the work of of M. S X We start by showing that g is continuous. On a Banach space X, the plank problem for polynomials consists in finding conditions on nonnegative scalars a 1, These constants have been studied by several authors.

In particular, this result implies the result of Arias-de-Reyna about the polarization constants mentioned eesigualdades. We also obtain some less restrictive conditions for some particular Banach spaces, like the L p spaces or Schatten classes S p.

We also give some estimates on polinomialss norm of the product of linear functions on l d Cthus obtaining bounds for the nth polarization constant c n l d C. Start display at page:. El factor problem consiste en buscar cotas inferiores para la norma del producto de polinomios de grados previamente fijados.