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Degree of pointedness of a convex function. (English) Zbl 0880.49016

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)

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

Citations:

Zbl 0193.18401
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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
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