On the Equal Hull Problem for Nontrivial Semiconvex Hulls

We define a nontrivial semiconvex hull

of a compact set

K\subset M^{N\times n} K ⊂ M N × n

called the

\alpha α

-rank-one convex quadratic hull and establish the equalities of semiconvex hulls with respect to

qr_{\alpha}(K) q r α ( K )

by showing that

L_c(K)=qr_{\alpha}(K) L c ( K ) = q r α ( K )

if and only if

Q(K)=qr_\alpha(K) Q ( K ) = q r α ( K )

,

0<\alpha<1 0 < α < 1

, where

Q(K) Q ( K )

and

L_c(K) L c ( K )

are the quasiconvex convex hull and the closed lamination convex hull of

K K

respectively. We also show that

qr_{\alpha}(K) q r α ( K )

is a nontrivial semiconvex hull, that is,

qr_{\alpha}(K)\neq C(K) q r α ( K ) ≠ C ( K )

if

R(K)\neq C(K) R ( K ) ≠ C ( K )