Bellman vs. Beurling: sharp estimates of uniform convexity for \(L^p\) spaces (Q2786984)

A Banach space \(X\) is said to be uniformly convex if for any \(\varepsilon>0\) there exists \(\delta>0\) such that for \(x,y\in X\) with \(\| x\|=\| y\|=1\) the condition \(\| x-y\|\geq\varepsilon\) implies that \(\|\frac{x-y}{2}\|\leq 1-\delta\). \textit{J. A. Clarkson} [Trans. Am. Math. Soc. 40, 396--414 (1936; Zbl 0015.35604)] introduced this notion and proved that all Lebesgue spaces \(L^p\) are uniformly convex for \(p\in(1,\infty)\). The modulus of convexity of a Banach space \(X\) is the function \(\delta:[0,2]\rightarrow [0,1]\) defined by NEWLINE\[NEWLINE\delta(\varepsilon)=\inf\left\{1-\left\|\frac{x-y}{2}\right\|:x,y\in X,\,\| x\|=\| y\|=1,\,\| x-y\|\geq\varepsilon\right\}.NEWLINE\]NEWLINE Using Clarkson's [loc. cit.] and \textit{O. Hanner}'s inequalities [Ark. Mat. 3, 239--244 (1956; Zbl 0071.32801)], the modulus of convexity can explicitly be calculated for all Lebesgue spaces \(L^p\) with \(p \in (1,\infty)\). The authors provide another way to calculate these moduli of convexity using the Bellman function method.NEWLINENEWLINEFix \(p\in(1,\infty)\). For a fixed point \((x_1,x_2,x_3)\in\mathbb R^3\), consider the set NEWLINE\[NEWLINET(x_1,x_2,x_3)=\left\{(\varphi,\psi)\in L^p[0,1]\times L^p[0,1]:\, \|\varphi\|^p_p=x_1,\|\psi\|^p_p=x_2,\, \|\varphi-\psi\|^p_p=x_3\right\}NEWLINE\]NEWLINE and define the Bellman function \(\mathbf B\) by the formula NEWLINE\[NEWLINE\mathbf B(x_1, x_2)=\sup\left\{\|\varphi+\psi\|_p^p:(\varphi,\psi)\in T(x_1,x_2,1-x_1-x_2)\right\}.NEWLINE\]NEWLINE The natural domain of \(\mathbf B\) is the convex set \(\Omega=\left\{(x_1, x_2)\in\mathbb R^2:T(x_1,x_2,1-x_1-x_2)\neq\emptyset\right\}\). The authors show that \(\mathbf B\) is concave on \(\Omega\) and that it is minimal among the concave functions with fixed boundary values. They study the behavior of \(\mathbf B\) on the boundary of \(\Omega\) in detail and are then capable to deduce implicitly the values of \(\mathbf B\) on the intersection of \(\Omega\) and the diagonal \(\{(x_1,x_2)\in\mathbb R^2:x_1=x_2\}\) by using the concavity of \(\mathbf B\). These values are then enough to calculate the required modulus of convexity of \(L^p[0,1]\).