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Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form $$ z = \sum_{i=0}^{n-1} x_i \omega …

The keyword you are looking for is "quantitative isoperimetric inequalities". …

Here is a more problem solving approach, since this problem comes from AoPS. Instead of working with $f$, work with $g = f''$ which is a convex function. Note that it is enough to assume $g(1)=0$. In …

I am studying an article: The parametric problem of capillarity: the case of two and three fluids, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this: …

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} …

I have found an article which deals with this kind of inequalities. It is available in the following link: Funzioni BV e Tracce …

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ques …