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Minimal oscillation and vanishing of smooth functions
Part of:
Nonlinear functional analysis
Equations and inequalities involving nonlinear operators
Measures, integration, derivative, holomorphy
Published online by Cambridge University Press: 04 September 2025
Abstract
In his paper [2], Ricceri considers, for X bounded convex subset of the real Hilbert space 
$ H,$ the quantity where 
$$ \begin{align*}\delta_X = \inf_{\varphi\in \Gamma_X} \Big( \sup_{x\in X} \big(\kern-1.2pt\left\Vert x\right\Vert{}^2 +\varphi(x)\big) - \inf_{x\in X} \big(\kern-1.2pt\left\Vert x\right\Vert{}^2 +\varphi(x) \Big),\end{align*} $$
$ \Gamma _X$ denotes the set of real convex functions on X, and shows that 
$\delta _X>0$ for 
$ X$ non singleton without giving any quantitative estimation of this quantity. And he asks, whether 
$\delta _X$ can be controlled by a function of the diameter of 
$ X$.
In this article, we show that 
$\delta _X$ is exactly the square of the Chebyshev radius of 
$ X$, hence is at least 
$\dfrac {\operatorname {\mathrm {{diam}}}(X)^2}4$. We deduce from the main result of [2] a quantitative statement on the zeros of a 
$\mathcal C^1$-operator on 
$ H$ with Lipschitz derivative, and show that this statement is optimal.
MSC classification
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
References
Alimov, A. R. and Tsar’kov, I. G.,
Chebyshev centres, Jung constants, and their applications
. Russ. Math. Surv. 74(2019), 775–849.Google Scholar
Ricceri, B.,
A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range
. Canad. Math. Bull. 68(2025), no. 2, 395–400.10.4153/S0008439524000821CrossRefGoogle Scholar