Definition
The support function of a non-empty closed convex set ''A'' in is given by : ; see T. Bonnesen, W. Fenchel, '' Theorie der konvexen Körper,'' Julius Springer, Berlin, 1934. English translation: ''Theory of convex bodies,'' BCS Associates, Moscow, ID, 1987. R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006. .R. Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993. Its interpretation is most intuitive when ''x'' is a unit vector: by definition, ''A'' is contained in the closed half space : and there is at least one point of ''A'' in the boundary : of this half space. The hyperplane ''H''(''x'') is therefore called a ''supporting hyperplane'' with ''exterior'' (or ''outer'') unit normal vector ''x''. The word ''exterior'' is important here, as the orientation of ''x'' plays a role, the set ''H''(''x'') is in general different from ''H''(-''x''). Now ''h''''A'' is the (signed) distance of ''H''(''x'') from the origin.Examples
The support function of a singleton ''A''= is . The support function of the Euclidean unit ball ''B''''1'' is . If ''A'' is a line segment through the origin with endpoints -''a'' and ''a'' then .Properties
As a function of ''x''
The support function of a ''compact'' nonempty convex set is real valued and continuous, but if the set is closed and unbounded, its support function is extended real valued (it takes the value ). As any nonempty closed convex set is the intersection of its supporting half spaces, the function ''h''''A'' determines ''A'' uniquely. This can be used to describe certain geometric properties of convex sets analytically. For instance, a set ''A'' is point symmetric with respect to the origin if and only if ''h''''A'' is an even function. In general, the support function is not differentiable. However, directional derivatives exist and yield support functions of support sets. If ''A'' is ''compact'' and convex, and ''h''''A'''(''u'';''x'') denotes the directional derivative of ''h''''A'' at ''u'' ≠ ''0'' in direction ''x'', we have : Here ''H''(''u'') is the supporting hyperplane of ''A'' with exterior normal vector ''u'', defined above. If ''A'' ∩ ''H''(''u'') is a singleton , say, it follows that the support function is differentiable at ''u'' and its gradient coincides with ''y''. Conversely, if ''h''''A'' is differentiable at ''u'', then ''A'' ∩ ''H''(''u'') is a singleton. Hence ''h''''A'' is differentiable at all points ''u'' ≠ ''0'' if and only if ''A'' is ''strictly convex'' (the boundary of ''A'' does not contain any line segments). It follows directly from its definition that the support function is positive homogeneous: : and subadditive: : It follows that ''h''''A'' is a convex function. It is crucial in convex geometry that these properties characterize support functions: Any positive homogeneous, convex, real valued function on is the support function of a nonempty compact convex set. Several proofs are known , one is using the fact that the Legendre transform of a positive homogeneous, convex, real valued function is the (convex) indicator function of a compact convex set. Many authors restrict the support function to the Euclidean unit sphere and consider it as a function on ''S''''n''-1. The homogeneity property shows that this restriction determines the support function on , as defined above.As a function of ''A''
The support functions of a dilated or translated set are closely related to the original set ''A'': : and : The latter generalises to : where ''A'' + ''B'' denotes theVariants
In contrast to the above, support functions are sometimes defined on the boundary of ''A'' rather than on ''S''''n''-1, under the assumption that there exists a unique exterior unit normal at each boundary point. Convexity is not needed for the definition. For an oriented regular surface, ''M'', with a unit normal vector, ''N'', defined everywhere on its surface, the support function is then defined by : . In other words, for any , this support function gives the signed distance of the unique hyperplane that touches ''M'' in ''x''.See also
*References
{{reflist Convex geometry Types of functions