Convexity and Legendre transform

Question 1

convex set

Answer 1

for all x,y in set and λ in [0,1]
λx+(1-λ)y in set

Question 2

convex function

Answer 2

on a convex set
f(λx+(1-λ)y)<=λf(x)+(1-λ)f(y)

Question 3

strictly convex function

Answer 3

f(λx+(1-λ)y)<λf(x)+(1-λ)f(y)

Question 4

non-negative definite matrix

Answer 4

<x,Ax»=0

Question 5

positive definite matrix

Answer 5

<x,Ax»0

Question 6

Legendre transform

Answer 6

f(x)=sup{<x*,x>-f(x):x in R^n}

Question 7

coercive function

Answer 7

f(x)/|x| tends to infinity as x tends to infinity