The algorithm requires the projection to be bounded by a convex polygon with six or eight vertices.
Then one has the inequalities f ( a + b 2, c + d 2 ) ≤ 1 ( b − a ) ( d − c ) ∫ a b ∫ c d f ( x, y ) d y d x ≤ f ( a, c ) + f ( a, d ) + f ( b, c ) + f ( b, d ) 4. Now, let △ be a convex area from the plane R 2, bounded by a convex function y = h ( x ) and a concave function y = g ( x ) and x = a, x = b, such that for any x ∈ [ a, b ], g ( x ) ≥ h ( x ).
For comparative and other reasons enumerated below, bounds on the innermost 95% of the data are also used to estimate the areas of HRs even for methods of construction that are able to produce HRs bounded by a 100% isopleth of a UD (e.g. the minimum convex polygon MCP, bounded parametric kernel methods, our LoCoH methods).
Theorem 3.3 Let △ be a bounded area by a convex function h and a concave function g on [ a, b ] such that for any x ∈ [ a, b ], g ( x ) ≥ h ( x ).
Theorem 3.2 Let △ be a bounded area by a convex function y = h ( x ) and a concave function y = g ( x ) on [ a, b ] such that for any x ∈ [ a, b ], g ( x ) ≥ h ( x ) and g − h is increasing on [ a, b ].
Suppose that all nuclei are located in an area which is bounded by two convex surfaces (e.g., in a spherical layer).
Taken together, in a crystal with a single planar defect, the fast addition of metal to the concave sides can cause those very faces to grow out of existence, leading to a triangular plate whose side faces are bounded by three convex sides that do not favor atomic addition.
We then show how the POCS estimate can be upper bounded by solving a non-convex optimization problem.
Because the containing region is a convex polygon bounded by circular arcs, the containing circle provides a reasonable approximation for the objective of our application.
In two dimensions, the expression reduces to a function of only one angle (theta =arctan {frac{n_y}{n_x}}), and the corresponding Wulff shape is a convex hull (Mullins 1962) bounded by the perpendiculars to (varvec{n}), at a distance (sigma (varvec{n})), from the origin (Sekerka 2005).
We prove the negative result that convex measures are bounded by O(n2) and show that several measures considered for proving lower bounds on the formula size are convex.
