Problems for next week

There is no seminar on Monday, but here are some problems for you.

• Show from the definition of convexity that if a convex function has a maximum, then there must be one on the boundary of its domain of definition.
  Is it true that there cannot be an interior maximum?
   
• If a strictly convex f is defined on the entire real line, can it then possibly be bounded?
   
• From this old problem set, do Problem 1 first bullet point, Problem 2, Problem 5 and Problem 6.
   
• Define f(t) = (2 + R + R² + R³)|t| + |t - 1| + R|t - 2| + R²|t - 3| + R³|t - 4| + t²⁰¹⁴ 
  and g(x) = f(||x||). Here, t is a single real variable, while x is a vector of arbitrary dimension n.
  Show that f is convex with (strict) minimum at zero, and decide whether g is convex or concave or neither. (Does the latter depend on n?)