Exercise · ZW-MINKOWSKI-INEQUALITY-010 · ★★★ Advanced
For \(0<p<1\), which norm property fails for \(\|\cdot\|_p\)?
Choose one option.
Show answer
- D. The triangle inequality
Explanation
For \(0<p<1\) positivity and homogeneity still hold, but the triangle inequality fails, as the example with \(p=1/2\) shows.
Review the concept: Minkowski inequality
Inequality stating that the \(L^p\) norm of a sum of functions is at most the sum of their \(L^p\) norms, for \(p \ge 1\).
\[\|f+g\|_p \le \|f\|_p + \|g\|_p\]
More practice problems: minkowski inequality
- ZW-MINKOWSKI-INEQUALITY-001 What is the Minkowski inequality for p ≥ 1?
- ZW-MINKOWSKI-INEQUALITY-002 For which values of p does the Minkowski inequality hold?
- ZW-MINKOWSKI-INEQUALITY-003 Which basic real-number inequality is the Minkowski inequality related to?
- ZW-MINKOWSKI-INEQUALITY-004 If p = 1, what does the Minkowski inequality state?
- ZW-MINKOWSKI-INEQUALITY-005 When does equality hold in the Minkowski inequality for 1<p<∞?
- ZW-MINKOWSKI-INEQUALITY-006 With f(x)=x on [0,1] and p=2, what is \|f\|_2?
- ZW-MINKOWSKI-INEQUALITY-007 With f(x)=x, g(x)=1 on [0,1] and p=2, what is \|f+g\|_2?
- ZW-MINKOWSKI-INEQUALITY-008 For f(x)=x and g(x)=1 on [0,1] with p=2, does the Minkowski inequality hold?