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The cavetto moulding is the opposite of the convex, bulging, ovolo, which is equally common in the tradition of Western classical architecture. Both bring the surface forward, and are often combined with other elements of moulding. Usually they include a curve through about a quarter-circle (90°). [3]
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
A convex quadrilateral is tangential if and only if opposite sides have equal sums. Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle. Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an ex-tangential quadrilateral. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential. [16]
A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, but connects alternating vertices. For an n-sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as {n/m}.
Every concave function is quasiconcave, but the opposite is not true. Every monotone transformation of a quasiconcave function is also quasiconcave. For example, if f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is quasiconcave and g : R → R {\displaystyle g:\mathbb {R} \to \mathbb {R} } is a monotonically-increasing ...
The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear maps . In particular, if C {\displaystyle C} is a convex cone, so is its opposite − C {\displaystyle -C} , and C ∩ − C {\displaystyle C\cap -C} is the ...