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On convexity properties of homogeneous functions of degree one. (English) Zbl 0895.49002
The authors give an explicit example of a function that is homogeneous of degree one, rank-one convex, but not convex. The function is defined by $$f(\xi)=|\xi|+\gamma\langle M\xi;\xi\rangle/| \xi|$$ if $$\xi\not= 0$$, $$f(0)=0$$. Here $$\xi\in R^{2\times 2}$$, $$| \xi|$$ denotes the Euclidean norm of $$\xi$$ in $$R^4$$, $$\langle\cdot;\cdot\rangle$$ is the scalar product in $$R^4$$, $$\gamma\geq 0$$ is appropriately chosen to produce a rank-one function which is not convex, and $$M$$ is a symmetric matrix, which commutes with the matrix $$E$$ ($$E$$ is the symmetric matrix defined by $$\text{det }\xi={1\over 2}\langle E\xi;\xi\rangle$$). The following choice $$M=\left(\begin{matrix} 9&0&0&1\\ 0&6&2&0\\ 0&2&6&0\\ 1&0&0&9\end{matrix}\right)$$, and $$\gamma\in ({1\over 2},1)$$ gives an explicit counterexample. It is derived from a more general theorem which relies the eigenvalues of $$M$$ and $$\gamma$$, with the convexity and the rank-one convexity properties of $$f$$.

##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 26B25 Convexity of real functions of several variables, generalizations 49J45 Methods involving semicontinuity and convergence; relaxation 49K10 Optimality conditions for free problems in two or more independent variables
##### Keywords:
rank-one convexity; quasiconvexity; convexity
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##### References:
  Sverak, Proc. Roy. Soc. Edinburgh Sect. A 120 pp 185– (1992) · Zbl 0777.49015  Dacorogna, Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001  Dacorogna, Progress in partial differential equations, Pont-à-Mousson, 1994 2 (1995)  DOI: 10.1512/iumj.1992.41.41017 · Zbl 0736.26006
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