Prove that a subspace of R^n is a convex set. Show that a rectangle in R^n is a

Question

Prove that a subspace of R^n is a convex set. Show that a rectangle in R^n is a convex set. Let H_2 be the half-space in R^n defined by c^T x greaterthanorequalto k. Show that H_2 is convex. Show that a hyperplane H in R^n is convex (Theorem 1.2). Show that the intersection of a finite collection of convex sets is convex (Theorem 1.3). Give two proofs of Theorem 1.4. One proof should use the definition of convex sets and the other should use Theorem 1.3. Consider the linear programming problem Maximize z = c^T x subject to Ax lessthanorequalto b x greaterthanorequalto 0. Let x_1 and x_2 be feasible solutions to the problem. Show that, if the objective function has the value k at both x_1 and x_2, then it has the value k at any point on the line segment joining x_1 and x_2. Show that the set of all solutions to Ax lessthanorequalto b, if it is nonempty, is a convex set Show that the set of solutions to Ax > b, if it is nonempty, is a convex set.

Explanation / Answer

32) Let x1 and x2 be feasible solutions of the problem

Then k = cTx1 and k = cTx2

Consider cTtx1 + cT(1-t)x2 = t(cTx1 ) + (1-t)(cTx2); Here 0<t<1

=> cTtx1 + cT(1-t)x2 = tk + (1-t)k = tk + k-tk = k

Hence tx1 + (1-t)x2 is also solution of cTx = k

So Any point on line joining x1 and x2 also has value of objective function equals k

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