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On moduli of \(k\)-convexity. (English) Zbl 0985.46002

If \(X\) is a Banach space with unit ball \(B_X\), the classical modulus of convexity [J. A. Clarkson, Trans. Am. Math. Soc. 40, 396-414 (1936; Zbl 0015.35604)] is defined by: \[ \delta_X(\varepsilon) := \inf \{ 1 - \|x_0 + x_1\|/2 : x_0,x_1 \in B_X, \|x_1 - x_0\|\geq \varepsilon\}. \] V. I. Gurarij [Mat. Issled. 2, No. 1(3), 141-148 (1967; Zbl 0232.46024)] showed that \(\delta_X(\varepsilon)\) is continuous on \([0,2)\) but not necessarily at 2. The notion of \(k\)-convexity was introduced by F. Sullivan [Can. J. Math. 31, 628-636 (1979; Zbl 0422.46011)] as follows. First let \[ A(x_0, x_1, \ldots ,x_k) := (k!)^{-1}\sup \{ \det (f_i(x_j - x_0))\} \] where the supremum is taken over all linear functionals \(f_1,\ldots, f_k\) with norm \(\leq 1\). This represents a “volume” of \(\text{co}(x_0,x_1, \ldots, x_k)\) and generalizes \(\|x_1 - x_0\|\). Now define \[ \delta_X^k(\varepsilon) := \inf \{ 1 - \|x_0 + x_1+ \cdots +x_k\|/(k+1) : x_0,x_1, \ldots ,x_k \in B_X, A(x_0, x_1, \ldots x_k) \geq \varepsilon)\}. \] In this paper the author constructs a delicate computation to show that \(\delta_X^k(\varepsilon)\) is continuous on \([0, \mu_X^k)\) where \(\mu_X^k:= \sup A(x_0,x_1,\ldots,x_k)\) where the supremum is over all \((k+1)\)-tuples from \(B_X\). This answers a question posed by W. A. Kirk [Proc. Symp. Pure Math., Am. Math. Soc. 45/2, 51-64 (1986; Zbl 0594.47048)].

MSC:

46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
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