Minimal oscillation and vanishing of smooth functions

Abstract In his paper [2], Ricceri considers, for X bounded convex subset of the real Hilbert space $ H,$ the quantity $$ \begin{align*}\delta_X = \inf_{\varphi\in \Gamma_X} \Big( \sup_{x\in X} \big(\kern-1.2pt\left\Vert x\right\Vert{}^2 +\varphi(x)\big) - \inf_{x\in X} \big(\kern-1.2pt\left\Vert x\right\Vert{}^2 +\varphi(x) \Big),\end{align*} $$ where $ \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.