The optimal problems for torsional rigidity

: In this paper, we consider the optimization problems associated with the nonhomogeneous and homogeneous Orlicz mixed torsional rigidities by investigating the properties of the corresponding mixed torsional rigidity. As the main results, the existence and the continuity of the solutions to these problems are proved.


Introduction
The setting of this paper is in the n-dimensional Euclidean space R n with inner product •, • .A set of points K in R n is convex if for all x, y ∈ K satisfying [x, y] ⊆ K.If C, D are compact convex sets in R n and λ ≥ 0, the Minkowski sum of C and D is C + D = {x + y : x ∈ C, y ∈ D}, and the scalar product λC is λC = {λx : x ∈ C}.Let K and K 0 be the class of convex bodies (compact convex set with nonempty interior) and the class of convex bodies which contain the origin o in their interiors, respectively.
The variation of volume of Minkowski sum of K ∈ K and the unit ball B n 2 ⊂ R n is the classical Borel measure, that is, the surface area of the convex body K can be formulated as: where |K| is the volume of K.More generally, for a fixed convex body Q, the relative surface area of a convex body K (relative to a convex body Q) can be given by is taking over on K, then Petty (see [28]) considered the following optimization problem and provided the solution as follows: there exists a convex body M with |M where Q • is the polar body of Q defined by Q • = {x ∈ R n : x, y ≤ 1, for all y ∈ Q}.The minimum S (K, M) is called the geominimal surface area of K, denoted by G(K) = S (K, M) for geometric meaning, by Petty (see [28]).
The solutions M and M in Theorem 1.1 are called the Orlicz − Petty bodies f or torsional rigidity.We use the set P 1,ϕ (K) to denote the collection of convex bodies M, and the set P 1,ϕ (K) to denote the collection of convex bodies M. For simplicity, we write Since the solutions M and M are unique if ϕ is convex by Theorem 1.1, then the sets P 1,ϕ (K) and P 1,ϕ (K) contain only one element and thus define two operators, we still use P 1,ϕ (K) and P 1,ϕ (K) to denote these two operators.Thus the continuity of Q 1,ϕ (K), Q 1,ϕ (K), P 1,ϕ (K) and P 1,ϕ (K) can be obtained.Theorem 1.2.Let ϕ : (0, ∞) → (0, ∞) be strictly increasing function with i=1 ⊆ K 0 be a sequence that converges to K ∈ K 0 .Then, the following statements hold:

Background and preliminaries
and x, y ∈ K.A convex body is a convex compact subset of R n with nonempty interior.Let K and K 0 be the class of convex bodies and the class of convex bodies with the origin in their interiors, respectively.For K, L ∈ K, denoted by K + L, the Minkowski sum, is defined as K + L = {x + y : x ∈ K, y ∈ L}.The scalar product of α ∈ R and K ∈ K, denote by αK, is defined as αK = {αx : x ∈ K}.For K ∈ K, |K| denotes to the volume of K and |B n 2 | = ω n denotes the volume of the unit ball B n 2 in R n .For K ∈ K, the volume radius of K, is defined as .
For any K ∈ K 0 , the surface area measure S (K, •) of K (see [1]), is defined as follows: where ν −1 K : S n−1 → ∂K (where ∂ denotes the boundary) is the inverse Gauss map and H n−1 is the (n − 1)-dimensional Hausdorff measure on ∂K.
Let C(S n−1 ) be the class of all continuous functions on S n−1 .The following two Lemmas will be useful: i=1 on S n−1 converges weakly to a finite measure µ on S n−1 and a sequence of functions Next we will introduce some basic concepts about the torsional rigidity which can be found in [6,16].Suppose C ∞ c (R n ) is the class of all infinitely differentiable functions on R n with compact supports.The torsional rigidity of a convex body K, denote by τ(K), is defined as (see [5]): where ∇u is the gradient of u and W 1,2 (intK) (where intK is the interior of K) is appropriate for the Sobolev space of the functions in L 2 (intK) whose first-order weak derivatives belong to L 2 (intK), and (2.2) The torsional rigidity is positively homogeneity of degree n + 2, that is, τ(aK) = a n+2 τ(K), for any K ∈ K 0 and a > 0. The torsional measure µ τ (K, •) is a nonnegative Borel measure on S n−1 which can be defined as (see [6]): for any measurable subset A ⊆ S n−1 , (2.4) For any a > 0, it is easy to check that In addition, µ τ (K, •) is not concentrated on any closed hemisphere of S n−1 , that is, where v, u + = max{ v, u , 0}.
From the previous definition (2.1) and (2.4), we have the following relation between µ τ (K, •) and S (K, •) as follows: By using the previous Borel measure, the integral formula of torsional rigidity τ was provided by Colesanti and Fimiani (see [6]) as follows: suppose K ∈ K with h K being the support function, then where u is the solution of (2.3).For any K ∈ K 0 , by (2.6), denote µ * τ (K, •) by a probability measure on (2.7)
Definition 3.1.Let ϕ ∈ I ∪ D and K, L ∈ K 0 .The Orlicz L ϕ mixed torsional rigidity of K and L, denoted by τ 1,ϕ (K, L), is defined as Clearly, τ 1,ϕ (K, K) = τ(K) for any ϕ ∈ I ∪ D. In addition, for the particular example of the previous definition, it is easy to verify that for any c > 0. Thus τ 1,ϕ (•, •) is nonhomogeneous if ϕ is nonhomogeneous.In this section, we introduce the homogeneous Orlicz L ϕ mixed torsional rigidity as follows.
•) uniformly on S n−1 .These further imply that there exist r, R > 0 with r ≤ R such that and Together with the continuity of ϕ, we have The convergence K i → K, also yields that Combining with Lemma 2.1, we have 2 ) by (2.2).Together with (3.2) and (3.3), we have i.e., τ 1,ϕ (K i , L i ) is bounded from above and below.Let So, there exist two subsequences In the same manner, one can check that Combing (3.4) and (3.5), we have This, together with the fact that " lim inf ≤ lim sup ", yields that τ 1,ϕ (K i , L i ) → τ 1,ϕ (K, L) as i → ∞.As for ϕ ∈ I, it can be obtained in the same way.
This implies that there exist n 0 ∈ N and a constant c 0 > 0 such that where Ω = {u ∈ S n−1 : u, v + ≥ 1 n 0 }.In addition, there exist two numbers r 0 , R 0 > 0 with r 0 ≤ R 0 such that by the compactness of S n−1 .Since ϕ ∈ I is increasing, Definition 3.1 and Lemma 2.1, we have that for any constant T > 0, Letting T → ∞, then c ≥ ∞.This is a contradiction, which shows that {M i } ∞ i=1 is uniformly bounded.Along the same line, one can check that {M i } ∞ i=1 is uniformly bounded when { τ 1,ϕ (K i , M i )} ∞ i=1 is bounded.

The Orlicz-Petty bodies for torsional rigidity
In this section, we will prove the existence, uniqueness and continuity of the Orlicz-Petty bodies for torsional rigidity.To do so, we study the following optimization problems for nonhomogeneous and homogeneous Orlicz L ϕ mixed torsional rigidities: The next theorem gives the existence of the solutions to the problems in (4.1) and (4.2).
Theorem 4.1.Suppose that K ∈ K 0 and ϕ ∈ I.The following statements hold: (i) There exists a convex body M ∈ K 0 with |M • | = ω n and (ii) There exists a convex body M ∈ K 0 with | M • | = ω n and Moreover, both of M and M are unique if ϕ ∈ I is convex.
Proof.For simplicity, we write 3) and Definition 3.1, we have i=1 is uniformly bounded by Theorem 3.2.This together with Lemma 2.2, we have a subsequence Hence M is a solution to (4.1).(ii) By (4.4) and Definition 3.2, we have i=1 is uniformly bounded by Theorem 3.2.This together with Lemma 2.2, we have a subsequence The proofs of the uniqueness of M and M are similar, so we only provide the proof for M. Assume that M 1 , M 2 ∈ K 0 and M 1 , M 2 satisfy Let N = (M 1 + M 2 )/2, by the Brunn-Minkowski inequality, vrad(N • ) ≤ 1 with equality if and only if By the monotonicity and convexity of ϕ, one has

This shows that vrad(N
We call the solutions M and M Orlicz − Petty bodies f or torsional rigidity.Following the idea of Petty, we call the minimums ϕ (K, M) the corresponding geominimal sur f ace area f or torsional rigidity.We use P 1,ϕ (•) and P 1,ϕ (•) to denote the sets of M and M, respectively.Definition 4.1.Suppose that K ∈ K 0 and ϕ ∈ I. Define the set Analogously, define the set Obviously, the sets P 1,ϕ (K) and P 1,ϕ (K) are nonempty which follow from Theorem 4.1 if ϕ ∈ I. Since P 1,ϕ (K) and P 1,ϕ (K) contain one element if ϕ ∈ I is convex, P 1,ϕ : K 0 → K 0 and P 1,ϕ : K 0 → K 0 define two operators on K 0 .The next theorem shows the continuity of Q 1,ϕ (•), Q 1,ϕ (•), P 1,ϕ (•) and P 1,ϕ (•).Theorem 4.2.Let ϕ ∈ I and {K i } ∞ i=1 ⊆ K 0 and K ∈ K 0 be such that K i → K as i → ∞.The following statements hold: Since {M i k } ∞ k=1 is uniformly bounded, and by Lemma 2.2, there exist a subsequence Combining (4.5) with (4.6), we have Next, we prove that . By Theorem 3.1 and (4.4), we have
Since { M i l } ∞ l=1 is uniformly bounded, and by Lemma 2.2, there exists a subsequence From (4.8) and (4.9), one concludes that (ii) Assume that ϕ ∈ I is convex.By Theorem 4.1, P 1,ϕ (K), P 1,ϕ (K i ), P 1,ϕ (K) and P 1,ϕ (K i ) contain one element which will be denoted by M, M i , M and M i for i ≥ 1, respectively.
. By (4.7) and (4.10) It follows that M = S and M = I.That is, M i → M and M i → M as i → ∞.
The following proposition shows that the Orlicz-Petty bodies for torsional rigidity of polytopes are still polytopes.Proposition 4.1.If ϕ ∈ I and K ∈ K 0 is a polytope, then the elements in P 1,ϕ (K) and P 1,ϕ (K) are polytopes with faces parallel to those of K.
Proof.Since K is a polytope, then S (K, •) must be concentrated on a finite subset {u 1 , u 2 , . . ., u m } ⊆ S n−1 .By (2.5), the torsional measure µ τ (K, •) is also concentrated on {u 1 , u 2 , . . ., u m }.If M ∈ P 1,ϕ (K), then let P 1 be a polytope with {u 1 , u 2 , . . ., u m } as the unit normal vectors of its faces such that P By (4.3), we have Since ϕ is strictly increasing, then vrad(P K) is a polytope with faces parallel to those of K.
Using the same method, one can prove that each M ∈ P 1,ϕ (K) is a polytope with faces parallel to those of K.
Finally, we list some counterexamples to show that the problems (4.1) and (4.2) may not be solvable in general case.
Then there exists a constant b > 0 such that b j ≥ b for all 1 ≤ j ≤ n.Since K is a polytope with u 1 , u 2 , . . ., u m as the unit normal vectors of its faces, we know that K is bounded, then there exists a constant c > 0 such that h K (u i ) ≤ c for 1 ≤ i ≤ m.For any d > 0, we write where 1 is in the jth column of the matrix T d .Then, L .
Due to ϕ ∈ D is decreasing, so Similarly, we can check that sup{ τ 1,ϕ (K, L) : L ∈ K 0 and |L • | = ω n } = ∞ if ϕ ∈ D. (ii) Firstly, suppose that µ τ (K, {u 1 }) > 0. Since K ∈ K 0 , then there exists a positive number c 1 such that h K (u i ) ≥ c 1 > 0 as 1 ≤ i ≤ m.Since K is a polytope with u 1 , u 2 , . . ., u m as the unit normal vectors of its faces, then K is bounded, namely, there exists a constant c 0 > 0 such that h K (u i ) ≤ c 0 for 1 ≤ i ≤ m.By the Schmidt orthogonalization, it can be found an orthogonal matrix T ∈ O(n) with u 1 as its first column vector.For any d > 0, let Sinse ϕ is increasing, then Similarly, one can check that sup{ τ 1,ϕ (K, L) : L ∈ K 0 and |L • | = ω n } = ∞ under the condition that ϕ ∈ I.

Conclusions
In this paper, we introduce the definition of the homogeneous Orlicz mixed torsional rigidities and obtain some properties of the nonhomogeneous and homogeneous Orlicz mixed torsional rigidities.Then we consider the optimization problems about the corresponding mixed torsional rigidity.As the main results, we prove the existence and the continuity of the solutions to these problems.
d | = ω n and