| # Exercises Convexity > Exercises ## Convexity For both positive scalars, show that Justify carefully your approach. Use the above result to prove that the function defined as is concave. Show that for any numbers , we have Consider the set defined by the following inequalities Draw the set. Is it convex? Show that it can be described as a single quadratic inequality of the form , for matrix , and which you will determine. What is the convex hull of this set? Prove Young's inequality, valid for every non-negative numbers , and integers with , : Hint: use the convexity of the exponential function. ## Simple optimization problems Solve the optimization problems below: determine the optimal value and the optimal set. . subject to , , . subject to . subject to . subject to . Consider the optimization problem Make a sketch of the feasible set. For each of the following objective functions, give the optimal set and the optimal value. . . . .