×

On linear transformations of intersections. (English) Zbl 1460.52005

In this paper the authors study (necessary and sufficient) conditions for the linear transformation \(T\) of the intersection of two closed convex sets \(A,B\) to coincide with the intersection of the images. A first easy result shows that if \(A\cup B\) is convex, then \(T(A\cap B)=T(A)\cap T(B)\). Other conditions regarding convexity in a direction are also established.
Furthermore, the following characterization is proved: \(T(A\cap B)=T(A)\cap T(B)\) if and only if the set \(\{x\in A\cup B:Tx=t\}\) is connected for all \(t\in T(A\cup B)\). They also show that all these conditions can be extended to non-convex closed sets, continuous transformations and multiple sets.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A35 Helly-type theorems and geometric transversal theory
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Aumann, RJ; Hart, S., Bi-convexity and bi-martingales, Israel J. Math., 54, 2, 159-180 (1986) · Zbl 0607.52001
[2] Axler, SJ, Linear Algebra Done Right (1997), New York: Springer, New York · Zbl 0886.15001
[3] Breen, M., Starshaped unions and nonempty intersections of convex sets in \(\mathbb{R}^d\) ℝd, Proc. Amer. Math. Soc., 108, 3, 817-820 (1990) · Zbl 0691.52006
[4] Bressan, A., Directional convexity and finite optimality conditions, J. Math. Anal. Appl., 125, 234-246 (1987) · Zbl 0627.49008
[5] Gershkov, A.; Goeree, JK; Kushnir, A.; Moldovanu, B.; Shi, X., On the equivalence of Bayesian and dominant strategy implementation, Econometrica, 81, 197-220 (2013) · Zbl 1274.91175
[6] Goeree, J.K., Kushnir, A.: A Geometric Approach to Mechanism Design Working Paper, University of New South Wales and Tepper School of Business, Carnegie Mellon University (2017)
[7] Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer Science & Business Media (2012) · Zbl 0998.49001
[8] Holtzman, JM; Halkin, H., Directional convexity and the maximum principle for discrete systems, SIAM J. Control, 4, 2, 263-275 (1966) · Zbl 0152.09302
[9] Horvath, C.; Lassonde, M., Intersection of sets with -connected unions, Proc. Amer. Math. Soc., 125, 4, 1209-1214 (1997) · Zbl 0861.54020
[10] Kushnir, A., On sufficiency of dominant strategy implementation in environments with correlated types, Econ. Lett., 133, 4-6 (2015) · Zbl 1364.91065
[11] Kushnir, A.; Liu, S., On the equivalence of Bayesian and dominant strategy implementation for environments with nonlinear utilities, Econ. Theory, 67, 617-644 (2019) · Zbl 1422.91140
[12] Laffont, J-J; Maskin, E., A differential approach to dominant strategy mechanisms, Econometrica, 48, 1507-1520 (1980) · Zbl 0443.90008
[13] Manelli, AM; Vincent, DR, Bayesian and dominant-strategy implementation in the independent private values model, Econometrica, 78, 1905-1938 (2010) · Zbl 1204.91057
[14] Milgrom, P.R.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004)
[15] Munkres, JR, Topology (2000), USA: Prentice Hall Inc., USA
[16] Myerson, R., Incentive compatibility and the bargaining problem, Econometrica, 47, 61-73 (1979) · Zbl 0399.90008
[17] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997) · Zbl 0932.90001
[18] Roth, AE, What have we learned from market design?, Econ. J., 118, 527, 285-310 (2008)
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.