Seeger, Alberto Degree of pointedness of a convex function. (English) Zbl 0880.49016 Bull. Aust. Math. Soc. 53, No. 1, 159-167 (1996). For a real-valued convex function \(f\) from \(\mathbb{R}^d\) to \(\mathbb{R}\), the author considers the recession cone \((\text{epi }f)_\infty\) of the epigraph \(\text{epi }f\) of \(f\) [see R. T. Rockafellar, “Convex analysis” (1970; Zbl 0193.18401)] and introduces as a measure of “pointedness” the functional \[ p[f]:=d-\dim l((\text{epi }f)_\infty)\text{ (``degree of pointedness'')}, \] where \(l(K)\) is defined as \(K\cap-K\) for a cone \(K\). He then shows that \(p[f]=\dim(\text{dom }f^*)\), where \(f^*\) denotes the effective domain of the conjugate function \(f^*\) of \(f\). The functional \(p(f)\) can also be expressed as the maximum of the dimension of convex subsets \(A\) of \(\mathbb{R}^d\) such that \(f-\psi^*_A\) is bounded from below by some constant where \(\psi^*_A\) denotes the support function of \(A\). Next, the author derives inequalities of the type \(p[f]\geq\max\{p[f_k]\mid 1\leq k\leq n\}\), where \(f\) is the maximum of the convex functions \(f_k\) or their sum. The methods used are close to those used by Rockafellar. Reviewer: B.Kind (Bochum) Cited in 2 Documents MSC: 49J52 Nonsmooth analysis 52A41 Convex functions and convex programs in convex geometry 49N15 Duality theory (optimization) 26B25 Convexity of real functions of several variables, generalizations Keywords:Fenchel conjugate function; degree of pointedness; convex function; epigraph Citations:Zbl 0193.18401 PDFBibTeX XMLCite \textit{A. Seeger}, Bull. Aust. Math. Soc. 53, No. 1, 159--167 (1996; Zbl 0880.49016) Full Text: DOI References: [1] DOI: 10.1007/BF00250669 · Zbl 0411.49012 · doi:10.1007/BF00250669 [2] Singer, Bull. Austral. Math. Soc. 29 pp 193– (1979) [3] Benoist, C.R. Acad. Sci. Paris. Sér. I 316 pp 233– (1993) [4] Laurent, Approximation et optimisation (1972) [5] Rockafellar, Convex analysis (1970) · Zbl 0932.90001 · doi:10.1515/9781400873173 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.