Properties
For every proper convex function f on Rn there exist some b in Rn and β in R such that
for every x.
The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets and are convex sets in the vector space X, then the indicator functions and are proper convex functions, but is not convex (unless is convex), and is possibly identically equal to if (i.e. are complementary halfspaces).
The infimal convolution of two proper convex functions is convex but not necessarily proper convex.
Read more about this topic: Proper Convex Function
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)