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Sharp convex Lorentz-Sobolev inequalities. (English) Zbl 1220.26020

The Lorentz-Sobolev inequality has a long history. The geometric analogue of the \(L^1\)-Sobolev inequality is the Euclidean isoperimetric inequality.
New sharp Lorentz-Sobolev inequalities are obtained by convexifying level sets in Lorentz integrals via the \(L^p\)-Minkowski problem. New \(L^p\)-isocapacitary and isoperimetric inequalities are proved for Lipschitz star bodies. It is shown that the sharp convex Lorentz-Sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities. Finally, the authors pose an open question as follows:
Are the geometric inequality
\[ \Phi_p(K)\leq \left(\frac{p-1}{n-p}\right)^{p-1} C_p(K) \]
for all \( K\in K_0^n\) and the analytic inequality
\[ \int^\infty_0 \big(\langle f\rangle_t\big)\,dt\leq \left(\frac{p-1}{n-p}\right)^{p-1} \int^\infty_0 C_p\big(\langle f\rangle_t\big)\,dt \]
for all \( f\in C^{\infty}_{0}(\mathbb R^{n})\) true for \(1<p<n\)?
The paper is useful to budding researchers in this field.

MSC:

26D20 Other analytical inequalities
46E99 Linear function spaces and their duals
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