 Radially open convex set

ID: radially-open-convex-set

C

of a real vector space is radially open when, for each

x \in C

and each vector

v

, the points

x + t v

remain in

C

for all sufficiently small real

t

. No ambient topology is required. If

C

is also convex and contains zero, it is absorbing and its Minkowski functional

p_{C} (x) = in f {s > 0 : x \in s C}

is finite and sublinear, with

C = {p_{C} < 1}

. Convexity gives subadditivity; radial openness permits a slight dilation of any point of

C

, giving the strict inequality.

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